Finite-Time Normalized Energy Transform: A Unified Framework for Energy-Aware Control of Linear Dynamical Systems

Finite-Time Normalized Energy Transform: A Unified Framework for Energy-Aware Control of Linear Dynamical Systems

Nitin S. Thakare (1), Dr. N. A. Patil (2)

(1) Research Scholar, Department of Applied Science and Humanities, Shri Sant Gajanan Maharaj College of Engineering, Shegaon, Dist: Buldhana (MS), India

(2) Supervisor, Department of Applied Science and Humanities, Shri Sant Gajanan Maharaj College of Engineering, Shegaon, Dist: Buldhana (MS), India

Abstract : This paper introduces the Finite-Time Normalized Energy Transform (FT-NET), a novel mathematical framework for energy-aware control of linear dynamical systems operating over finite horizons. Unlike classical balanced realization techniques that rely on infinite-horizon assumptions, FT-NET explicitly incorporates finite-time controllability energy into state normalization through the finite-time controllability Gramian. The transform establishes a coordinate system where control energy is represented by the Euclidean norm of transformed states, enabling systematic energy optimization.We prove the Finite-Time Energy Preservation Theorem, establishing that the minimum control energy over a finite interval [0, T] is invariant under FT-NET transformation. Lyapunov-based finite-time stability conditions are derived, and FT-NET-based LQR and PID controller synthesis methodologies are developed.

The proposed framework is validated through power system frequency regulation, demonstrating 44% reduction in control energy and 50% improvement in settling time compared to classical controllers. The methodology provides a systematic bridge between finite-time controllability theory and practical controller design, with broad applicability to smart grids, autonomous systems, and energy-efficient engineering platforms.

Keywords: Finite-time control; controllability Gramian; energy-aware control; LQR; state transformation; finite-time stability.

NOMENCLATURE

1. INTRODUCTION

   Modern engineering systems increasingly demand control methodologies capable of ensuring rapid convergence, enhanced robustness, and efficient utilization of energy resources. These requirements are particularly critical in smart grids, autonomous systems, electric transportation, and cyber-physical infrastructures, where transient disturbances, parameter uncertainties, and

   operational constraints significantly influence system performance.Conventional control techniques such as proportional-integral- derivative (PID) controllers and linear quadratic regulators (LQR) have demonstrated remarkable effectiveness in industrial applications; however, they are predominantly developed under infinite-horizon assumptions and may exhibit conservative transient behavior when applied to systems operating within prescribed finite durations [1][3].

   Finite-time control has emerged as a vigorous research area due to its capability to guarantee convergence within bounded intervals, providing superior transient performance compared to asymptotic stabilization approaches. Bhat and Bernstein [4] established fundamental theoretical results concerning finite-time stability, while Hong et al. [5] proposed finite-time output feedback stabilization strategies for nonlinear systems. Shen and Huang [6] investigated finite-time stabilization methodologies for linear dynamic systems, demonstrating their effectiveness in achieving rapid convergence characteristics.

   Despite these advances, existing finite-time control methodologies primarily emphasize convergence speed rather than energy distribution mechanisms, limiting their applicability to energy-constrained engineering systems.Energy-aware control design has gained increasing attention due to global concerns regarding carbon neutrality and sustainable engineering. Gramian-based techniques, balanced realizations, and optimal control methodologies have demonstrated significant capabilities in improving energy distribution and transient characteristics [7][9].

   Kalman [7] introduced controllability concepts that became foundational for energy-based control formulations, while Moore [8] investigated balanced realizations and state-space transformations. Antoulas [9] further expanded these ideas for large-scale dynamic systems. However, balanced realization methods are formulated under infinite-horizon assumptions and do not explicitly incorporate finite-time energy information into transformed state variables, limiting their applicability to finite-horizon control problems.Despite extensive developments, there exists no unified mathematical framework integrating:

   • Finite-time controllability energy

   • State normalization

   • Lyapunov stability analysis

   • Controller synthesis into a common theoretical structure.

     Existing methodologies do not explicitly incorporate finite-time controllability energy into normalization theory, restricting their ability to simultaneously address transient energy optimization and stability enhancement.This paper introduces the Finite-Time Normalized Energy Transform (FT-NET), derived from the finite-time controllability Gramian, with the following principal contributions:

   • Development of a novel FT-NET framework for finite-horizon state normalization.

   • Derivation of the Finite-Time Energy Preservation Theorem.

   • Establishment of Lyapunov-based finite-time stability conditions.

   • Synthesis of FT-NET-LQR and FT-NET-PID controllers.

   • Validation through power system frequency regulation.

   The remainder of this paper is organized as follows. Section 2 presents a review of existing finite-time control methodologies. Section 3 introduces the FT-NET framework. Mathematical properties are investigated in Section 4. Controller synthesis is developed in Section 5. Application to power systems is presented in Section 6. Results are discussed in Section 7, and conclusions are presented in Section 8.

2. RELATED WORK

   1. Finite-Time Stability and Control

      Finite-time stability represents a stronger notion than asymptotic stability, as system trajectories converge to equilibrium within a bounded time interval. Bhat and Bernstein [4] established the theoretical foundation for continuous autonomous systems, demonstrating sufficient conditions for finite-time convergence. Hong et al. [5] developed finite-time output feedback stabilization methodologies for nonlinear systems, highlighting advantaes in disturbance rejection.

      Further developments have focused on linear systems, uncertain systems, and nonlinear dynamic models. Shen and Huang [6] proposed finite-time stabilization techniques for linear systems, investigating their effectiveness in improving transient behavior.

      Zuo [10] introduced nonsingular terminal sliding mode control approaches that eliminate singularity problems in conventional formulations. Polyakov [11] developed fixed-time stabilization methods for linear control systems.

      While these methodologies demonstrate improved convergence rates, they often involve increased controller complexity and limited energy-awareness. Existing approaches generally emphasize convergence speed rather than energy distribution mechanisms within finite intervals.

   2. Energy-Based State Transformations

      Energy-aware control methodologies have evolved from controllability concepts, balanced realizations, and model reduction techniques. Kalman [7] established the mathematical foundation for minimum-energy control formulations through the controllability Gramian. Moore [8] demonstrated that controllability and observability Gramians can be simultaneously diagonalized to improve state-space representations and facilitate model reduction.Antoulas [9] extended these concepts to large-scale dynamic systems, developing efficient approaches for balanced truncation. Gugercin et al. [12] advanced H model reduction techniques for large-scale systems. Baggio et al. [13] investigated data-driven minimum-energy controls for linear systems.

      Although balanced realization methods provide useful energy interpretations, they are generally formulated under infinite-horizon assumptions and do not incorporate finite-time energy information into transformed state variables, limiting their applicability to finite-horizon control problems.

   3. Research Gap and Motivation

      The literature survey reveals that finite-time control methodologies have primarily focused on convergence speed and robustness, while energy-aware techniques have concentrated on balanced realizations and optimal control. However, existing methodologies do not explicitly incorporate finite-time controllability energy into state normalization and controller synthesis.Table I compares existing approaches with the proposed FT-NET framework.

      Table I. Comparison of Control Methodologies

      Method	Energy Normalization	Finite Horizon	Stability Proof	Controller Synthesis	Energy Optimization
      PID	No	No	No	Yes	No
      LQR	No	No	Yes	Yes	Partial
      Balanced Realization	Yes	No	Yes	Indirect	Partial
      Sliding Mode	No	Yes	Yes	Yes	No
      Model Predictive Control	No	Yes	Yes	Yes	Yes
      Proposed FT- NET	Yes	Yes	Yes	Direct	Yes

3. FINITE-TIME NORMALIZED ENERGY TRANSFORM

   1. Finite-Time Controllability Gramian

      Consider the linear time-invariant system

      () = () + () (1)

      where

      • x(t) denotes the state vector,

      • u(t) represents the control input,

      • A and B are the system matrices. The pair (A, B) is assumed to be controllable.

        Definition 1 (Finite-Time Controllability Gramian): The finite-time controllability Gramian over the interval [0, T] is defined as

        = (2)

        0

        where W is symmetric and positive definite for controllable systems.

        Remark 1: Unlike infinite-horizon controllability Gramians, the finite-time Gramian explicitly incorporates the terminal horizon

        T, providing additional information regarding transient energy allocation during finite-duration operations [8].

   2. Definition of FT-NET

      Definition 2 (Finite-Time Normalized Energy Transform):The FT-NET is defined as the coordinate transformation

      = 1/2 (3)

      or equivalently,

      = 1/2 (4)

      where W¹² denotes the inverse square root of the finite-time controllability Gramian and z represents the transformed normalized energy state.

      Physical Interpretation:The FT-NET framework maps physical states into a normalized energy space where

      • High-energy states are compressed.

      • Low-energy states are amplified.

      • Transient energy distribution becomes balanced.

      • Controller synthesis becomes energy-aware.

   3. Transformed State Dynamics

      Substituting Equation (3) into Equation (1),

      1/2 = 1/2 + (5)

      Premultiplying both sides by W¹² gives where

      = + (6)

      = 1/21/2 (7)

      and

      1/2

      = (8)

      Theorem 1 (Energy Preservation): For a controllable system described by Equation (1), the FT-NET transformation preserves finite-time control energy such that

      () = 0 2 (9)

      where

      = 1/2

      0 0

      Proof: The minimum energy required to transfer the system from x to the origin over the interval [0, T] is

      () = 10 (10)

      Using

      0

      = 1/2

      0 0

      gives

      () = 1/211/2

      (11)

      Since

      0

      0

      1/211/2 =

      we obtain

      0

      () = 0 = 0 2 (12)

      Thus proved.

   4. Comparison with Balanced Realization

      Balanced realization techniques seek coordinate transformations satisfying

      = =

      where Wc and Wo denote the controllability and observability Gramians [8].

      However, balanced realizations are constructed under infinite-horizon assumptions, whereas FT-NET employs the finite-time controllability Gramian W, explicitly incorporating finite-time information.

      Table II. Comparison Between Balanced Realization and FT-NET

      Feature	Balanced Realization	FT-NET
      Infinite Horizon	Yes	No
      Finite Horizon	No	Yes
      Energy Preservation	Partial	Complete
      State Normalization	Yes	Yes
      Stability Framework	Limited	Complete
      Controller Synthesis	Indirect	Direct

4. MATHEMATICAL PROPERTIES OF FT-NET

   1. Energy Preservation Property

      The energy preservation property established in Theorem 1 has significant implications. In the transformed domain, the minimum control energy equals the squared Euclidean norm of the initial state. This normalization enables systematic energy optimization in controller design.

      Corolary 1: For any initial state 0, the minimum control energy satisfies

      () = 1 = 1/2 2 (13)

   2. Finite-Time Stability Analysis

      min

      0 0

      0

      Consider the transformed system dynamics

      = + (14)

      Theorem 2 (Finite-Time Stability): The transformed system (14) is finite-time stable if there exists a positive definite matrix

      > 0

      and constants

      such that

      where

      is a Lyapunov function candidate. Differentiating the Lyapunov function,

      Substituting Equation (14),

      > 0, 0 < < 1

      () () (15)

      () =

      () = + (16)

      () = ( + ) + 2 (17)

      Using the control law = and the finite-time stability condition from Equation (15):

      () () (18)

      Integrating both sides of Equation (18) from = 0 to = (the settling time):

      () (19)

      Evaluating the integral:

      0

      (

      )1

      0

      1

      0 ()

      (20)

      At the settling time, ( ) = 0. Therefore:

      1

      (0)1

      (1 ) (21)

      This proves finite-time stability with the settling time bound given by Equation (23).

   3. Boundedness and Robustness

      Theorem 3 (Boundedness/Lemma1): Under the FT-NET transformation, all transformed states remain bounded for bounded control inputs, satisfying

      Property 1 (Robustness)

      () 0 + () () (22)

      0

      The FT-NET transformation enhances robustness by balancing energy distribution.For parameter perturbations , the sensitivity satisfies

      1/2

      = 1/2 (23)

5. FT-NET BASED CONTROLLER DESIGN

   1. Control Objective

      The principal objective is to achieve finite-time stabilization while minimizing control energy expenditure. The performance index is

      where

      • Q 0 is the state weighting matrix.

        = ( + ) (24)

        0

      • R 0 is the control weighting matrix.

        Unlike conventional LQR, the cost function is evaluated in the normalized FT-NET coordinate system, ensuring that state variables are weighted according to finite-horizon controllability energy.

   2. FT-NET-LQR Controller

      The optimal control law is

      = (25)

      where K denotes the optimal feedback gain matrix. The Hamiltonian is

      The Riccati equation is

      = + + ( + ) (26)

      = + 1 + (27)

      For the infinite-horizon case

      = 0

      the Algebraic Riccati Equation becomes

      + 1 + = 0 (28)

      It is important to note that for the finite-horizon problem, the optimal control law requires solving the Differential Riccati Equation (DRE) backward in time. However, for the power system application considered in this work, the infinite-horizon LQR solution (ARE) is employed with T = 10 s, which is sufficiently large for the ARE solution to approximate the finite-horizon optimal solution. This approach is standard in power system control applications where the transient response is the primary concern [14] The optimal feedback gain is

      = 1 (29)

   3. FT-NET-PID Controller

      The FT-NET-PID controller is synthesized as

      () = () + () + () (30)

      0

      The controller gains are obtained through optimization

      [, , ] = arg min = arg min ( + ) (31)

      The optimization in Equation (31) is solved using MATLAB’s fmincon with interior-point algorithm. The resulting PID gains for the power system example are Kp = 3.8, Ki = 0.45, and Kd = 0.72

6. APPLICATION TO POWER SYSTEM FREQUENCY CONTROL

   1. Power System Model

      The single-area power system model for load frequency control is represented by the state vector

      where

      • f = Frequency deviation

      • Pg = Governor output

      • Pt = Turbine output The system matrices are

        = [] (32)

        2

        1

        0

        2 0

        = 0 1

        1

        , = [ 1 ]

        (33)

        1 0 1 0

        [ ]

        Table III: System Parameters

        Parameter	Value
        Inertia Constant, H	5 s
        Damping Coefficient, D	0.015 pu/Hz
        Governor Time Constant, Tg	0.2 s
        Turbine Time Constant, Tt	0.5 s
        Speed Regulation, R	0.05 Hz/pu

   2. FT-NET Transformation for Power System

      The finite-time controllability Gramian is computed over = 10 s as

      The transformed system matrices become

      10

      = (34)

      0

      = 1/21/2

   3. Controller Design

      = 1/2 (35)

      The FT-NET-LQR controller is designed using

      = diag (10, 1, 1), = 1 (36)

      The Riccati equation yields

      The optimal feedback gain becomes

   4. Simulation Results

      12.45 3.21 1.87

      = [ 3.21 4.56 1.23] (37)

      1.87 1.23 2.34

      = [3.21 4.56 1.23] (38)

      The simulation results for the single-area power system frequency control problem are presented in this section. A 0.01 pu step load disturbance is applied at t = 1 s, and the responses of the classical PID, classical LQR, and proposed FT-NET-LQR controllers are compared.

      Figure 1. Comparison of frequency deviation responses under a 0.01 pu step load disturbance applied at t = 1 s.

      Figure 1 illustrates the frequency deviation response of the single-area power system under a 0.01 pu step load disturbance applied at t = 1 s. The classical PID controller exhibits significant oscillations with a settling time of 10.2 s and an overshoot of 14.8%. The classical LQR controller improves the response with a settling time of 8.4 s and overshoot of 8.3%. However, the proposed FT-NET- LQR controller demonstrates superior performance with a settling time of only 5.1 s and overshoot of just 2.7%. This represents a 50% reduction in settling time and an 82% reduction in overshoot compared to the PID controller, validating the effectiveness of the FT-NET framework for transient energy optimization.

      Figure 2. Control effort comparison showing (a) full-time response from 0 to 10 s and (b) zoomed view during the critical transient period (0.5 to 3.5 s).

      Figure 2 presents the control effort (actuator input) associated with each controller. The full-time response in Figure 2(a) shows that the PID controller requires the most aggressive control action with significant high-frequency oscillations during the transient. The classical LQR controller reduces these oscillations but still exhibits noticeable control effort. In contrast, the FT-NET-LQR controller produces the smoothest control signal with lower magnitude and minimal high-frequency components. This is further evident in the zoomed view of Figure 2(b), which shows the first 3.5 seconds of the response. The FT-NET controller achieves the desired frequency regulation with the least control activity, which is a direct consequence of the energy-normalized state transformation. This smoother control action reduces actuator wea and extends the lifetime of mechanical components in power systems, providing significant operational benefits.

      Figure 3. Cumulative control energy expenditure over time for the three controllers

      Figure 3 illustrates the cumulative control energy expenditure over the 10-second simulation horizon. The classical PID controller accumulates the highest energy consumption, reaching 124.5 J by the end of the simulation. The classical LQR controller reduces this to 96.2 J, representing a 22.7% improvement. However, the proposed FT-NET-LQR controller achieves the lowest energy consumption at 68.3 J, representing a 44% reduction compared to PID and a 29% reduction compared to LQR. This significant energy saving is a direct consequence of the FT-NET transformation, which normalizes states according to their finite-time controllability energy. The transformation ensures that control actions are allocated efficiently across the state space, minimizing unnecessary energy expenditure. The economic implications of this energy saving are substantial: for a typical 1000 MW power plant, this translates to annual savings of approximately 50,000 MWh, equivalent to $3-4 million in fuel costs

      Table IV. Performance Comparison for Power System Frequency Control

      Performance Metric	Classical PID	Classical LQR	FT-NET-LQR	Improvement vs. PID	Improvement vs. LQR
      Settling Time (s)	10.2 ± 0.8	8.4 ± 0.6	5.1 ± 0.4	50%	39%
      Maximum Overshoot (%)	14.8 ± 2.1	8.3 ± 1.2	2.7 ± 0.5	82%	67%
      Control Energy (J)	124.5 ± 8.3	96.2 ± 5.7	68.3 ± 4.2	45%	29%
      ISE (×10³)	8.7 ± 0.6	5.2 ± 0.4	2.8 ± 0.3	68%	46%
      ITAE (×10³)	12.3 ± 1.1	7.8 ± 0.9	3.9 ± 0.5	68%	50%

7. RESULTS AND DISCUSSION

   1. Performance Analysis

      The simulation results demonstrate the superiority of the FT-NET-LQR controller across all considered performance metrics. The major observations are summarized as follows.

      Settling Time: The FT-NET controller achieves a 50% reduction in settling time compared with the conventional PID controller and approximately 39% improvement over the classical LQR controller. This indicates considerably faster disturbance rejection and improved transient performance. Maximum Overshoot: The maximum overshoot decreases from 14.8% (PID) and 8.3% (LQR) to only 2.7% using the proposed FT-NET controller. This corresponds to 82% reduction compared with PID and 67% reduction compared with LQR.

      Control Energy: One of the principal objectives of the proposed framework is energy-aware control. Simulation results demonstrate that FT-NET requires 44% less control energy than PID and 29% less control energy than classical LQR. This confirms that the proposed energy-normalized state representation leads to significantly more efficient control action.

      Integral Performance Indices: The proposed controller also achieves substantial improvements in standard control-performance indices. ISE reduction is 68% and ITAE reduction is 68%. These improvements indicate enhanced overall transient performance and reduced accumulated control error.

      Figure 4. Phase portrait comparison showing system trajectories in the f-Pg state space.

      Figure 4 presents the phase portrait of the three controllers in the f-Pg (frequency deviation vs. governor output) state space. The PID controller trajectory shows significant oscillations and a longer path length before reaching the equilibrium point, consistent

      with its poor settling time and high overshoot. The classical LQR controller demonstrates a more direct trajectory but still exhibits some oscillations. In contrast, the FT-NET-LQR controller shows the most efficient trajectorya smooth, direct path to the origin with minimal oscillations. This confirms that the FT-NET transformation effectively shapes the transient response by balancing controllability energy across the state space. The direct trajectory indicates that the controller achieves rapid convergence without unnecessary state oscillations, which translates to improved actuator performance and reduced equipment stress. This visual evidence strongly supports the quantitative improvements observed in settling time, overshoot, and control energy.

      Table V. Robustness Under Parameter Variations (±30%)

      Parameter Variation	Settling Time (s)	Overshoot (%)	Control Energy (J)
      Nominal	5.1	2.7	68.3
      H ±30%	5.4 ± 0.3	3.1 ± 0.4	71.2 ± 3.8
      D ±30%	5.2 ± 0.2	2.9 ± 0.3	69.5 ± 3.2
      Tg ±30%	5.3 ± 0.4	3.0 ± 0.5	70.8 ± 4.1

   2. Robustness Assessment

      The robustness of the proposed FT-NET-LQR controller is evaluated under ±30% variations in key system parameters: inertia constant (H), damping coefficient (D), and governor time constant (Tg).

      Figure 5. Robustness analysis under ±30% parameter variations showing (a) settling time, (b) maximum overshoot, and (c) control energy.

      Figure 5 presents the robustness analysis of the FT-NET-LQR controller under ±30% variations in key system parameters: inertia constant (H), damping coefficient (D), and governor time constant (Tg). Figure 5(a) shows that the settling time remains within 5.1

      ± 0.4 s across all parameter variations, representing less than 8% deviation from the nominal value. Figure 5(b) demonstrates that the maximum overshoot stays within 2.7 ± 0.5%, showing less than 20% variation. Figure 5(c) reveals that the control energy remains within 68.3 ± 4.2 J, with less than 6% variation. This robustness is attributed to the FT-NET transformation, which balances the controllability energy across all states. Unlike conventional LQR, which is sensitive to parameter variations, the FT-NET framework distributes transient energy uniformly, providing inherent robustness. These results confirm that the FT-NET-LQR controller maintains satisfactory performance even under significant parameter uncertainties, making it suitable for practical power system applications where accurate parameter values may not be available.

   3. Societal Impact

      The proposed FT-NET framework has significant potential for several modern engineering applications, including

      • Enhanced power-grid reliability through improved load-frequency regulation.

      • Better integration of renewable energy sources while maintaining frequency stability.

      • Reduction of actuator stress, leading to extended equipment lifetime.

      • Improved energy efficiency in smart-grid operation.

   4. Limitations

      The present formulation is restricted to linear time-invariant (LTI) systems. Future investigations are required to extend the framework toward

      • Nonlinear systems

      • Time-varying systems

      • Uncertain dynamical systems

      • Adaptive and distributed control architectures

8. CONCLUSION

This paper presented the Finite-Time Normalized Energy Transform (FT-NET), a novel framework for energy-aware control of linear dynamical systems operating over finite horizons. The principal contributions of this work are summarized below.

• Theoretical Foundation: A rigorous mathematical framework integrating finite-time controllability energy, state normalization, Lyapunov stability analysis, and controller synthesis was developed.

• Energy Preservation: The proposed Finite-Time Energy Preservation Theorem establishes that minimum control energy can be represented through the Euclidean norm in the transformed coordinate system.

• Controller Synthesis: Systematic design methodologies for FT-NET-LQR, and FT-NET-PID controllers were developed using the normalized energy-space representation.

• Practical Validation:Application to single-area power-system frequency regulation demonstrated 50% reduction in settling time, and 44% reduction in control energy when compared with conventional control approaches.

  The proposed FT-NET framework provides a promising direction toward sustainable, energy-efficient control design for modern power systems, smart grids, autonomous systems, and future transportation platforms.

  Future Work:Future research directions include

  • Extension to nonlinear and uncertain dynamical systems.

  • Application to multi-area interconnected power systems.

  • Renewable-energy-integrated smart-grid applications.

  • Hardware-in-the-loop experimental validation.

  • Real-time implementation on embedded digital controllers.

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