Maximum Power Control of Hybrid Wind-Diesel-Storage System

Abstract

Extraction of maximum wind power of variable speed wind turbines in hybrid wind-diesel-storage system (HWDSS) is considered due to economical purposes. The proposed control algorithm utilizes extended fuzzy-linear matrix equalities (FLMEs) systems design of stabilizing fuzzy controllers for nonlinear systems described by Takagi-Sugeno (TS) fuzzy models. The algorithm maximizes the power coefficient for a fixed pitch. Moreover, it reduces the voltage ripple and stabilizes the system over a wide range of wind speed variations. The control scheme is tested for different profiles of wind speed pattern and provides satisfactory results.

1. Introduction

Since ancient times, wind turbine technology has been used to improve humankind’s quality of life where people have used wind turbines to pump water and mill grain, along with many other uses. Today, wind turbines are used for similar purposes (i.e., water or oil pumping, battery charging, or utility generation) as a cheap, clean source of electricity, and well suited for isolated places with no connections to the electric grid [1].

In remote areas such as small islands, diesel generators are the main power supply. Diesel fuel has several drawbacks: it is quite expensive because transportation to remote areas adds extracost, and it causes pollution via engine exhaust. Providing a feasible, economical, and environmental alternative source to diesel generators is important. A hybrid wind-diesel system with battery storage of wind power can benefit islands and other isolated communities and increase fuel savings. However, control of such system is very important as wind turbine produces excessive fluctuation of power output, which negatively influences the quality of electricity supplied to the load, particularly frequency and voltage [2].

Optimum wind energy extraction is achieved by running the wind turbine generator (WTG) in variable-speed, variable-frequency mode. The rotor speed is allowed to vary in sympathy with the wind speed, by maintaining the tip speed ratio to a value that maximizes aerodynamic efficiency. In order to achieve this ratio, the permanent magnet synchronous generator load line should be matched very closely to the maximum power line of the wind turbine generator [3].

The problem of wind energy conversion system output power control has been considered extensively [4–7]. Maximization of the wind energy conversion efficiency based on a brushless doubly fed reluctance generator is discussed in [4]. Reference [5] maximizes power based on a standard V/Hz converter and controls the frequency to achieve the desired power at a given turbine speed. Reference [6] maximizes power based on controlling the slip power, which is extracted from the rotor circuits and fed to the grid though a rectifier-inverter branch. The firing angle of the inverter is used to control the slip power. Reference [7] presents a hill-climb searching (HCS) control for the maximum wind turbine power at variable wind speeds.

The main contribution of this research is to maximize the energy from the wind in the presence of a wide range of wind variations using the proposed FLME controller. Also, it provides a robust controller that stabilizes the HWDSS and overcomes the system nonlinearity. In addition, it guarantees good robustness and performance of the controller. Finally, it reduces the voltage ripple on the main bus voltage. The proposed FLME controller is based on the Takagi-Sugeno (TS) fuzzy model and linear matrix inequalities [8–10].

This paper is organized as follows. Section 2 provides system analysis. Section 3 presents the design of the proposed FLME controller. Section 4 shows the stability and robustness conditions for the proposed algorithm. Section 5 presents simulation of the wind turbine. Finally, concluding remarks are made in Section 6 followed by the list of references.

2. System Analysis

2.1. Wind Turbine Modeling

The kinetic energy of the wind due to its speed is captured by the turbine and is converted to mechanical energy. Along with the turbine, there is a generator present at the tower top which is coupled to the wind turbine by a shaft and often with a gear box. The generator converts mechanical energy of turbine to electrical energy and it feeds at demand point. The global scheme of a variable speed is displayed in Figure 1. The expression for aerodynamic power (𝑃_𝑎) captured by the wind turbine is given by the nonlinear expression [11] 𝑃_𝑎=0.5𝐶_𝑝(𝜆)𝜌𝜋𝑅²𝑉³, (1) where 𝜌 is the air density (kg⋅m^-3), 𝑅 is the rotor radius (m), 𝑉 is the wind speed (m⋅s^-1), and 𝐶_𝑝 is the power coefficient defined by the following relation [12]: 𝐶_𝑝=(0.44−0.0167𝛽)sin[𝜋(𝜆−3)15−0.3𝛽]−0.00184(𝜆−3)𝛽, (2) where 𝛽 is the blade pitch angle of the wind turbine, 𝜆 is the tip speed ratio (TSR) and is given by [11] 𝜔𝜆=^𝑡𝑅𝑉, (3) where 𝜔_𝑡 is the rotational speed of the blades.

Figure 1 

Wind turbine system.

Referring to (2), optimal TSR 𝜆_opt can be obtained as follows: 𝜆_opt=(15−0.3𝛽𝜋)cos⁻¹[0.00184𝛽(15−0.3𝛽)𝜋(0.44−0.167𝛽)]+3. (4) Thus, the maximum power captured from the wind is given by 𝑃_𝑎(max)=0.5𝐶_𝑝(max)(𝜆_opt,𝛽)𝜌𝜋𝑅²𝑉³. (5)

A typical 𝐶_𝑝−𝜆 curve is shown in Figure 2. It can be seen that there is a maximum power coefficient 𝐶_𝑝(max). Normally, a variable speed wind turbine follows the 𝐶_𝑝(max) to capture the maximum power up to the rated speed by varying the rotor speed to keep the system at 𝜆_opt, then operates at the rated power with power control during the periods of high wind by the active control of the blade pitch angle or the passive regulation based on aerodynamic stall. A typical power-wind speed curve is shown in Figure 3.

Figure 2 

Power coefficient 𝐶𝑝 versus TSR 𝜆.

Figure 3 

Power-wind speed characteristics.

2.2. System Description

The underlying hybrid wind-diesel-storage system is illustrated in Figure 4. The hybrid generation system is composed of a wind turbine coupled with a synchronous generator, a diesel-induction generator, and an energy storage system. In the given system, the wind turbine drives the synchronous generator that operates in parallel with the storage battery system. When the wind generator alone provides sufficient power for the load, the diesel engine is disconnected from the induction generator. The PEI connecting the load to the main bus is used to fit the frequency of the power supplying the load as well as the voltage.

Figure 4 

Structural diagram of hybrid wind-diesel-storage system.

The dynamics of the system can be characterized by the following equations [2]: 𝑥̇=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑦=𝐶𝑥(𝑡), (6) where 𝑥(𝑡)=𝑉_𝑏𝜔_𝑠^𝑇,𝑢(𝑡)=𝐸_fd𝐼_ref^𝑇,𝐿𝐴=1101^𝑓𝜏^′_do𝜔_𝑠𝐿_md𝐿_𝑓𝜏^′_do𝐿_md𝜔𝑠(𝐿_𝑑𝑖_{sd−(𝑟𝑎𝑖sq/𝜔𝑠)})𝑃_ind−𝑃_ind𝐽_𝑠𝜔_𝑠𝑉_𝑏−𝐷_𝑠𝐽_𝑠,𝑉𝐵=1−^𝑐𝐽_𝑠𝜔_𝑠𝑉0−^𝑐𝐽_𝑠𝜔_𝑠,𝐶=,1001 (7) where 𝑉_𝑐 is the AC side line-to-line voltage, 𝐸_fd is the SG field voltage, 𝜔_𝑠 is the bus frequency (or angular speed of SG), 𝐽_𝑠  and  𝐷_𝑠 are the inertia and frictional damping of SG, 𝑖_sd  and  𝑖_sq are the direct and quadrature current components of SG, 𝐿_𝑑  and  𝐿_𝑓 are the stator d-axis and rotor inductance of SG, 𝐿_md is the d-axis field mutual inductance, 𝜏^′_do is the transient open circuit time constant, 𝑟_𝑎 is the rotor resistance of SG, 𝑃_ind is the power of the induction generator, 𝑃_load is the power of the load, 𝐼_ref is the direct current set point, and 𝑉_𝑏 is the bus voltage. Equation (6) indicates that the model is the linear form for fixed matrices A, B, and C. However, matrices A and B are not fixed, but change as functions of state variables, thus making the model nonlinear. Also, this model is only used as a tool for controller design purposes. The used system parameters are shown in Table 1 [13–15].

Table 1 

System parameters.

3. The Proposed FLME Controller

3.1. Takagi-Sugeno's Fuzzy Plant Model

The Takagi-Sugeno fuzzy model represents a nonlinear system by partitioning the system into subsystems and then combining them with linguistic rules. In this paper, three linear subsystems are considered for the nonlinear state-space models (6), The continuous fuzzy dynamic model, proposed by Takagi-Sugeno, is described by fuzzy IF-THEN rules, which represent local linear input-output relations of nonlinear systems [16]. The 𝑖th rule of this fuzzy model is given by Plant RuleIF𝑞₁(𝑥(𝑡))is𝑁^𝑖₁AND⋯AND𝑞_𝜓(𝑥(𝑡))is𝑁^𝑖_𝜓.Then𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),𝑦=𝐶_𝑖𝑥(𝑡), (8) where 𝑁^𝑖_Ω is a fuzzy set, Ω=1,2,…,𝜓, 𝑖=1,2,…,𝑝, 𝑥(𝑡)∈𝑅^𝑛×1 is the state vector, 𝑢(𝑡)∈𝑅^𝑛×1 is the input vector, 𝐴_𝑖∈𝑅^𝑛×𝑛 and 𝐵_𝑖∈𝑅^𝑛×𝑚 are system matrices of appropriate dimensions, 𝑝 is the number of IF-THEN rules (𝑝=3). 𝑞₁(𝑥(𝑡)),…,𝑞_𝜓(𝑥(𝑡)) are the premise variables. The plant dynamics are then described by 𝑥̇(𝑡)=^𝑝_𝑖=1ℎ_𝑖(𝑥(𝑡))[𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)], (9) where ℎ_𝑖Γ(𝑥(𝑡))=^𝑖(𝑥(𝑡))∑^𝑝_𝑖=1Γ_𝑖,Γ(𝑥(𝑡))_𝑖(𝑥(𝑡))=∏^𝜓_Ω=1𝑁^𝑖_Ω(𝑥(𝑡)),ℎ_𝑖>0,∑^𝑝_𝑖=1ℎ_𝑖(𝑥(𝑡))=1. (10)

3.2. Fuzzy Controller

Three controllers are designed for the three linear subsystems, and then the total control output is obtained by defuzzification. A state feedback by linear matrix equalities (LMEs) is used to design a controller for each subsystem. The control is performed so that the power coefficient is maximized, thus the maximum power captured from the wind is obtained.

The 𝑗th rule of fuzzy controller is given by Plant RuleIF𝑓₁(𝑥(𝑡))is𝑀^𝑗₁AND⋯AND𝑓_𝜓(𝑥(𝑡))is𝑀^𝑗_𝜓.Then𝑢(𝑡)=𝐺_𝑗𝑥(𝑡)+𝑟, (11) where 𝑀^𝑗_𝜙 is a fuzzy set 𝜙=1,2,…,𝜓, 𝑗=1,2,…,𝑐, 𝑟 is the reference input, 𝑓₁(𝑥(𝑡)),…,𝑓_𝜓(𝑥(𝑡)) are the premise variables, 𝑐 is the number of IF-THEN rules (𝑐=3), and 𝐺_𝑗 are local feedback gains. The inferred output of the fuzzy controller is given by 𝑢(𝑡)=^𝑐_𝑗=1𝑚_𝑗(𝑥(𝑡))[𝐺_𝑗𝑥(𝑡)+𝑟], (12) where 𝑚_𝑗𝜛(𝑥(𝑡))=^𝑗(𝑥(𝑡))∑^𝑐_𝑗=1𝜛_𝑗,𝜛(𝑥(𝑡))_𝑗(𝑥(𝑡))=∏^𝜓_𝜙𝑀^𝑗_𝜙,𝑚_𝑗>0,∑^𝑐_𝑗=1𝑚_𝑗=1. (13)

3.3. Parallel Design Approach (PDA)

It allows the stability criterion to be satisfied more easily. It is used when the membership functions are known and the same rule antecedents of the TS fuzzy plant model are used in the fuzzy controller. Referring to (9) and (12), the fuzzy control system is given by 𝑋̇(𝑡)=^𝑝𝑝_𝑖=1𝑖=1ℎ_𝑖(𝑥(𝑡))ℎ_𝑖(𝑥(𝑡))[𝐻_𝑖𝑖𝑋(𝑡)+𝐵_𝑖𝑟], (14) where 𝐻_𝑖𝑖=𝐴_𝑖+𝐵𝐺_𝑖.

For each subspace, different model (𝑖=𝑗=1,2,3), (𝑝=𝑐=1,2,3) is applied. The degree of membership function for states 𝑉_𝑏𝜓 and 𝜔_𝑠 is depicted in Figure 5. Each membership function also represents model uncertainty for each subsystem.

Figure 5 

Membership functions of states.

4. Stability and Robustness for the Proposed Algorithm

A proof of the stability and robustness conditions for the plant dynamics described by (9) is shown in the appendix. The main result is summarized in Lemma  1.

Lemma 1. Under PDA, the fuzzy control system as given by (14) is stable if 𝜇[𝑇𝐻_𝑖𝑖𝑇⁻¹]≤−𝜁, (15) where 𝜁 is nonzero positive constant and T is a transformation matrix. The analysis given in the appendix indicates that ‖𝑥(𝑡)‖ will go to its steady state faster if we use larger values of 𝜁. Calculation of 𝐺_𝑗 of the fuzzy controller that satisfies the stability and robustness conditions is formulated as an LME problem.
If 𝑇^𝑇=𝑇,𝑃=𝑇𝑇,𝐺_𝑗=−𝑅𝐵^𝑇_𝑗𝑃,∀𝑗, (16)𝑃>0,𝑅∈𝜅^𝑚×𝑚 are symmetric positive definite matrices. The transformation matrix (𝑇) should be found in such a way that the uncertainty free system is stable [17]. Using (14), 𝑃𝐻_𝑖𝑖+𝐻^𝑇_𝑖𝑖𝑃<0,𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃−2𝑥𝑃𝐵𝑅𝐵^𝑇𝑃=−𝜎𝐼𝑃>0,∀𝑖, (17) where 𝜎 is robustness index.

5. Simulation Results

The proposed controller for the HWDSS is tested for many cases of wind speed variations. Four wind speed signals are tested in this section to prove the effectiveness of the proposed algorithm.

5.1. Random Variation of Wind Speed Signal

In this case, the wind speed signal is considered as a random wave as shown in Figure 6. The rotor speed to capture the maximum power from the wind turbine is shown in Figure 7 (solid line). It is clear that the dotted curve in Figure 7 which represents the actual rotor speed coincides with the solid curve. As the wind speed ranges between the cut-in and rated speed of the wind turbine, the produced power curve takes almost the wind speed curve as shown in Figure 8. The power generated at wind speed of 12 m/s is 0.75 Mw. Comparing this value with that obtained using PI controller [14] which is 0.4 Mw, it is clear that a 45% increase is obtained in the maximum value. Figure 9 shows that the voltage profile is nearly constant and that the voltage ripple is reduced to 90% compared with the adaptive fuzzy logic control [15].

Figure 6 

Wind speed.

Figure 7 

Rotor speed tracking.

Figure 8 

Per unit wind turbine produced power.

Figure 9 

Bus voltage.

5.2. Sinusoidal Variation of Wind Speed Signal

In this case, the wind speed signal is considered as a sine wave as shown in Figure 10. The rotor speed to capture the maximum power from the wind turbine is shown in Figure 11 (solid line). It is clear that the dotted curve in Figure 11 which represents the actual rotor speed coincides with the solid curve. As the wind speed ranges between the cut-in and rated speed of the wind turbine, the produced power curve takes almost the wind speed curve as shown in Figure 12. Figure 13 shows that the voltage profile is nearly constant and the voltage ripple is reduced to 90% compared with [15].

Figure 10 

Wind speed.

Figure 11 

Rotor speed tracking.

Figure 12 

Per-unit wind turbine produced power.

Figure 13 

Bus voltage.

5.3. Step Change of Wind Speed

In this case, a big step change of wind speed is tested as shown in Figure 14. Figure 15 shows the rotor speed while the dotted line in this figure represents the actual rotor speed that the controller is able to capture. It is clear from this figure that the controller effectively tracks the rotor speeds. Figure 16 indicates the output power of the wind turbine as a per-unit value, it is clear that when the wind speed is below the cut-in speed or over cut-off speed, the power reaches zero. Figure 17 indicates that the bus voltage is nearly constant.

Figure 14 

Wind speed.

Figure 15 

Rotor speed tracking.

Figure 16 

Per-unit wind turbine produced power.

Figure 17 

Bus voltage.

5.4. Sinusoidal Variation of Wind Speed Signal

This case is another sinusoidal wind speed signal but its minimum values is 8 m/s and its maximum value is 26 m/s which is bigger than the cut-off speed of the turbine as shown in Figure 18. Figure 19 indicates the effectiveness of the algorithm which tracks the rotor speed for different wind speed, while Figure 20 shows the output power from the wind turbine, which indicates that when the speed become over the cut-off speed, the power equals zero. Figure 21 indicates that the bus voltage is nearly constant and voltage ripple reduced to 90% compared with [15], since the ripple voltage reduced to 0.045 V with the proposed algorithm, but in [15] the ripple voltage is 0.4 V.

Figure 18 

Wind speed.

Figure 19 

Rotor speed tracking.

Figure 20 

Per-unit wind turbine produced power.

Figure 21 

Bus voltage.

Comparing the results of the proposed algorithm with that given in [14, 15], it could be seen that the proposed controller has the following advantages.

(i)It can control the plant well over a wide range of wind speeds.(ii)The generated power is increased up to 45% compared with [14].(iii)The algorithm is more robust in the presence of high nonlinearity.(iv)Bus voltage is nearly constant and voltage ripple is reduced to 90% compared with [15].

6. Conclusion

This paper presents a hybrid power system consisted of a wind turbine, a diesel generation unit, and energy storage devices. Both the wind power generator and the SG operate at variable speeds so as to maximize the wind energy capture as a force source and minimize the diesel fuel consumption for economic purpose. Both types of generation units are connected to the ac load system through PEI to stabilize the system frequency. The control is performed so that the power coefficient is maximized. The operating principles have been discussed and the simulation model of the systems has been developed. The proposed algorithm utilizing FLME is simple and leads to robust control performance. Simulation results have confirmed that maximum power conversion efficiency obtained increases to the order of 45% compared with previous methods and voltage ripple reduced to 90%. Maximum power control of hybrid wind power generation with storage battery is achieved.

Appendix

Proof of The Stability and Robustness Conditions
Consider the Taylor series [17] 𝑥(𝑡+Δ𝑡)=𝑥(𝑡)+𝑥̇(𝑡)Δ𝑡+Φ(Δ𝑡), (A.1) where Φ(Δ𝑡)=𝑥(𝑡+Δ𝑡)−𝑥(𝑡)−𝑥̇(𝑡)Δ𝑡 is the error term and Δ𝑡>0 is lim_Δ𝑡→0⁺Φ(Δ𝑡)Δ𝑡=0. (A.2) From (14) and (A.1) and multiplying a transformation matrix 𝑇∈𝑅^𝑛×𝑛 of rank n to both sides and taking norm on both sides of the above equation, we have ‖𝑇(𝑥(𝑡+Δ𝑡))‖≤‖^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))(𝐼+𝑇𝐻_𝑖𝑗𝑇⁻¹Δ𝑡)‖‖𝑇𝑥(𝑡)‖+‖^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))[𝑇𝐵𝑟]Δ𝑡‖+‖𝑇Φ(Δ𝑡)‖, (A.3) where ‖⋅‖ denotes the L₂ norm for vectors and L₂ induced norm for matrices, from (A.3), lim_Δ𝑡→0⁺‖𝑇(𝑥(𝑡+Δ𝑡))‖−‖𝑇𝑥(𝑡)‖Δ𝑡≤lim_Δ𝑡→0⁺{𝒜(‖𝐼+𝑇𝐻_𝑖𝑗𝑇⁻¹Δ𝑡‖−1)‖𝑇𝑥(𝑡)‖}Δ𝑡+lim_Δ𝑡→0⁺{‖𝒜[𝑇𝐵𝑟]Δ𝑡‖+‖𝑇Φ(Δ𝑡)‖},Δ𝑡 (A.4)where 𝒜 denotes ∑^𝑝_𝑖=1∑^𝑐_𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡)). From (A.2) and (A.4), 𝑑‖𝑇𝑥(𝑡)‖≤𝑑𝑡^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))𝜇[𝑇𝐻_𝑖𝑗𝑇⁻¹]‖𝑇𝑥(𝑡)‖+‖^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))[𝑇𝐵𝑟]‖, (A.5) where 𝜇[𝑇𝐻_𝑖𝑗𝑇⁻¹]=lim_Δ𝑡→0⁺‖𝐼+𝑇𝐻_𝑖𝑗𝑇⁻¹Δ𝑡‖−1Δ𝑡=𝜂_max(𝑇𝐻_𝑖𝑗𝑇⁻¹+(𝑇𝐻_𝑖𝑗𝑇⁻¹)^∗2), (A.6) where 𝜂_max(⋅) is the largest eigenvalue, * is the conjugate transpose, from (A.5), 𝑑‖𝑇𝑥(𝑡)‖≤𝑑𝑡^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))(𝜇[𝑇𝐻_𝑖𝑗𝑇⁻¹])‖𝑇𝑥(𝑡)‖+‖^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))[𝑇𝐵𝑟]‖. (A.7) Let 𝜇[𝑇𝐻_𝑖𝑖𝑇⁻¹]≤−𝜁,∀𝑖, (A.8) from (A.7) and (A.8), 𝑑𝑑𝑡(‖𝑇𝑥(𝑡)‖𝑒^{𝜁(𝑡−𝑡0)})≤^𝑝𝑐_𝑖=1𝑗=1ℎ_𝑖(𝑥(𝑡))𝑚_𝑗(𝑥(𝑡))‖[𝑇𝐵𝑟]‖𝑒^{𝜁(𝑡−𝑡0)}, (A.9) where 𝑡₀<𝑡 is an arbitrary initial time, based on (A.9) there are two cases to investigate the system behavior. (1)𝑟=0,(2)𝑟≠0. (A.10) If condition (A.8) is satisfied, the closed loop system (14) is stable, and ‖𝑥(𝑡)‖→0, as 𝑡→∞.

Proof (1) (𝑟=0)
𝑑𝑑𝑡(‖𝑇𝑥(𝑡)‖𝑒^{𝜉(𝑡−𝑡0)})≤0‖𝑇𝑥(𝑡)‖≤‖𝑇𝑥(𝑡₀)‖𝑒^{−𝜉(𝑡−𝑡0)}. (A.11) Since 𝜉 is a positive value, ‖𝑥(𝑡)‖→0 as 𝑡→∞

Proof (2) (𝑟≠0)
From (A.9), (‖𝑇𝑥(𝑡)‖𝑒^{𝜉(𝑡−𝑡0)})≤‖𝑇𝑥(𝑡₀)‖+‖𝑇𝐵^Λ𝑟‖^𝑡_𝑡0𝑒^{𝜉(𝜏−𝑡0)}𝑑𝜏, (A.12) where ‖𝑇𝐵^Λ𝑟‖≤max_𝑖‖𝑇𝐵𝑟‖_max≤‖𝑇𝐵𝑟‖, (A.13) then ‖𝑇𝑥(𝑡)‖≤‖𝑇𝑥(𝑡₀)‖𝑒^{−𝜉(𝑡−𝑡0)}+‖𝑇𝐵^Λ𝑟‖𝜉(1−𝑒^{−𝜉(𝑡−𝑡0)}). (A.14)Since the right-hand side of (A.14) is finite if r is bounded, the system states (14) are also bounded.

The above analysis gives an upper bound of ‖𝑇𝑥(𝑡)‖ under the two different considered cases. The result is given by (A.11) and (A.14). Similarly, a lower bound of ‖𝑥(𝑡)‖ can be obtained by following the same analysis procedure with 𝑥(𝑡−Δ𝑡)=𝑥(𝑡)−𝑥̇(𝑡)Δ𝑡+𝜙(Δ𝑡), (A.15) where 𝜙(Δ𝑡)=𝑥(𝑡−Δ𝑡)−𝑥(𝑡)+𝑥̇(𝑡)Δ𝑡 is the error term and Δ𝑡>0 and 𝜗 is governed by the following

Let 𝜇[−𝑇𝐻_𝑖𝑖𝑇⁻¹]≤𝜗,∀𝑖, (A.16) since 𝜗 is a positive value.