Robust Control and Active Vibration Suppression in Dynamics of Smart Systems

: Challenging issues arise in the design of control strategies for piezoelectric smart structures. Piezoelectric materials have been investigated for use in distributed parameter systems in order to provide active control efﬁciently and affordably. In the active control of dynamic systems, distributed sensors and actuators can be created using piezoelectric materials. The three fundamental issues that structural control engineers must face when creating robust control laws are structural modeling methodologies, uncertainty modeling, and robustness validation. These issues are reviewed in this article. A smart structure with piezoelectric (PZT) materials is investigated for its active vibration response under dynamic disturbance. Numerical modeling with ﬁnite elements is used to achieve that. The vibration for different model values is presented considering the uncertainty of the modeling. A vibration suppression was achieved with a robust controller and with a reduced order controller. Results are presented for the frequency domain and the state space domain. This work cleary demostrated the advantage of robust control in the vibration suppration of smart stuctures.


Introduction
A piezoelectric structure with a control strategy has the potential to adapt to both a changing internal environment and a changing external environment, such as stresses or form changes.It includes intelligent actuators that enable controlled modification of system parameters and reactions.Piezoelectric materials (PZT), shape memory alloys, electrostrictive materials, magnetostrictive materials, and fiber optics are only a few examples of the numerous types of actuators and sensors under consideration.We employ piezoelectric material in our paper.In the active control of dynamic systems, piezoelectric materials can be specially adapted to serve as distributed sensors and actuators.The study of intelligent structures has drawn the attention of numerous scholars [1][2][3][4][5][6].A smart structure is one that keeps an eye on both its surroundings and itself [7,8].
Robust vibration control of piezoelectric-actuated smart structures has recently attracted a lot of attention.Despite the existence of numerous sources of uncertainty, such control laws are preferred for systems where guaranteed stability or performance are required [9][10][11].
The later robust controller accounts for the dynamical system's uncertainties as well as the incompleteness of the measured data, which results in the design of smart structures that can be used.To provide a thorough and unitary methodology for designing and validating reliable H infinity (H inf ) controllers for active structures, the numerical simulation demonstrates that sufficient vibration suppression can be achieved by using the suggested general methods in a tutorial manner for the case of a piezoelectric smart structure [12][13][14].The novelty of the work is that it calculates an H infinity controller with very good results in the frequency domain and the state space, even for different values of the mass and stiffness matrix, considering the uncertainty of the modeling; additionally, good results were acquired with a reduced order H infinity controller.No similar work achieves vibration suppression if there are different values of the mass and the stiffness matrix.

Materials and Methods
The approximate discretized variation problem results from using the traditional finite element method.By substituting discretized formulas into the initial variation of kinetic energy and strain energy for a finite element, discrete differential equations are generated [8,15].The beam element equation of motion is defined in terms of the nodal variable q as follows, integrating over spatial domains and applying Hamilton's principle [8,10] M ..
where K is the global stiffness matrix, D is the viscous damping matrix, M is the global mass matrix, f e is the global control force vector produced by electromechanical coupling effects, and f m is the global external loading vector for a beam structure used in this work.
Transversal deflections w i and rotations ψ i constitute the independent variable q(t), i.e., where in the analysis the number of finite elements used is the n index in the matrix.Vectors w and f m are upward positive.Permit state-space representation transformation of control (in the usual manner), Furthermore, to express f e (t) as Bu(t) we write it as F * e u where F * e (of size 2n × n) indicates the voltages on the actuators.The F * e (of size 2n × n) matrix also denotes the piezoelectric force for a unit mounted on its corresponding actuator.Lastly, the disturbance vector is designed by the following equation d(t) = f m (t) [15].Then, .
The output equation, as a function of the measured displacements, will help us to strengthen this, In the equation, the u parameter's matrix size is n × 1 (or smaller), while the d parameter's matrix size is 2n × 1.The units used are Newtons, radians, meters, and seconds.
In the next section, we will examine the behavior of a 32-element cantilever beam containing pairs of elements.The beam's dimensions are L × W × h.The sensors and actuators have a width and thickness of b S and b A , individually.The electromechanical properties of the beam of interest depicted in Figure 1a,b are listed in Table 1.In the next section, we will examine the behavior of a 32-element cantilever beam containing pairs of elements.The beam's dimensions are L × W × h.The sensors and actuators have a width and thickness of bS and bA, individually.The electromechanical properties of the beam of interest depicted in Figure 1a,b are listed in Table 1.

Frequency Domain
In a transfer function matrix, the structured singular value is defined as, (6) This matrix specifies the smallest structured Δ and has  (Δ) as a function (sigma is the structured singular value for the uncertainty modeling), and, as a result, the determinant becomes zero, i.e., det(I − MΔ) = 0: then () = 1/ (Δ).Equation (6) calculates the singular value.The upper and lower limits are visually presented and they should be less than one (1) for the specific Kp (arithmetic parameter for the stiffness matrix) and Km (arithmetic parameter for the scaled mass matrix) values.Following this, it is desired that the μ values are lower than 1, as shown in the results section.The principle followed was the smaller, the better [15][16][17].

Design Objectives
Design goals can be divided into two groups: Nominal performance 1.Small control effort.2. Attenuation of disturbances with acceptable transient characteristics (overshoot, settling time).

Frequency Domain
In a transfer function matrix, the structured singular value is defined as, This matrix specifies the smallest structured ∆ and has σ(∆) as a function (sigma is the structured singular value for the uncertainty modeling), and, as a result, the determinant becomes zero, i.e., det(I − M∆) = 0: then µ(M) = 1/σ(∆).Equation ( 6) calculates the singular value.The upper and lower limits are visually presented and they should be less than one (1) for the specific Kp (arithmetic parameter for the stiffness matrix) and Km (arithmetic parameter for the scaled mass matrix) values.Following this, it is desired that the µ values are lower than 1, as shown in the results section.The principle followed was the smaller, the better [15][16][17].

Design Objectives
Design goals can be divided into two groups: Nominal performance 1.
Small control effort.
Robust performance

4.
The above criteria ( 1)-( 3) should be satisfied even when noise exists in the modeling procedure.

System Specifications
To obtain the necessary system specifications, the system should be represented in the (N, ∆) structure to achieve the aforementioned objectives.The conventional diagram is depicted in Figure 2.

Strength of closed loop system (plant + controller).
Robust performance 4. The above criteria ( 1)-( 3) should be satisfied even when noise exists in the modeling procedure.

System Specifications
To obtain the necessary system specifications, the system should be represented in the (N, Δ) structure to achieve the aforementioned objectives.The conventional diagram is depicted in Figure 2. The disturbance vector (mechanical force) d and noise vector n are the diagram's two inputs, and the control vector u and state vector x are the diagram's two outputs.It is expected in what follows that, If that is not the case, then the original signals can be modified using the right frequency-dependent weights to give the altered signals this feature [9,13].
Rewrite Figure 2 similarly to Figure 3: Or with fewer details (Figure 4), The disturbance vector (mechanical force) d and noise vector n are the diagram's two inputs, and the control vector u and state vector x are the diagram's two outputs.It is expected in what follows that, If that is not the case, then the original signals can be modified using the right frequency-dependent weights to give the altered signals this feature [9,13].
Robust performance 4. The above criteria ( 1)-( 3) should be satisfied even when noise exists in the modeling procedure.

System Specifications
To obtain the necessary system specifications, the system should be represented in the (N, Δ) structure to achieve the aforementioned objectives.The conventional diagram is depicted in Figure 2.
If that is not the case, then the original signals can be modified using the right frequency-dependent weights to give the altered signals this feature [9,13].
Rewrite Figure 2 similarly to Figure 3: Or with fewer details (Figure 4), Or with fewer details (Figure 4),  with, where z is the output (control vector u, and the state vector x) controllable variables as well as exogenous inputs (mechanical disturbances vector and the noise) [12,14,18].Given that P is composed of two inputs and two outputs, it is typically partitioned as follows, with, where z is the output (control vector u, and the state vector x) controllable variables as well as exogenous inputs (mechanical disturbances vector and the noise) [12,14,18].Given that P is composed of two inputs and two outputs, it is typically partitioned as follows, z(s) y(s) = P zw (s) P zu (s) P yw (s) P yu (s) Also, u(s) = K s (s)y(s) (10) The transfer function for a closed loop is obtained by substituting (10) in ( 9) N zw (s) with K s (s) the controller of our system, N zw (s) = P zw (s) + P zu (s)K s (s)(I − P yu (s)K s (s)) To determine robustness prerequisites, an additional graph is needed, as shown in Figure 5: with, where z is the output (control vector u, and the state vector x) controllable variables as well as exogenous inputs (mechanical disturbances vector and the noise) [12,14,18].Given that P is composed of two inputs and two outputs, it is typically partitioned as follows,  (9) Also, u(s) = Ks(s)y(s) (10) The transfer function for a closed loop is obtained by substituting (10) in ( 9) Nzw(s) with Ks(s) the controller of our system, To determine robustness prerequisites, an additional graph is needed, as shown in Figure 5: where the N factor is defined by Equation ( 11) and the uncertainty parameter, which is modeled in Δ, should satisfy the following criterion ║Δ║∞ ≤ 1 (details later).Where We can state the following definitions based on this structure, shown in Table 2:  where the N factor is defined by Equation ( 11) and the uncertainty parameter, which is modeled in ∆, should satisfy the following criterion ||∆||∞ ≤ 1 (details later).Where We can state the following definitions based on this structure, shown in Table 2: The following conditions are demonstrated to be true for real or complex blockdiagonal perturbations ∆: If M is internally stable, the system is presumably stable; II.If the system performs about average; III.If and only if, the system (M, ∆) is robustly stable, sup ω∈R µ ∆ (N 11 (jω)) < 1 (13) where the structured singular value of N is the parameter µ ∆ in the criterion, for the structured uncertainty set ∆.This condition is known as the generalized small gain theorem [12][13][14].
IV.The system (N, ∆) exhibits robust performance if and only if, where and ∆ p is fully complex and has the same structure as ∆ and dimensions corresponding to (w, z).Unfortunately, only bounds on µ can be estimated [19,20].

Controller Synthesis
All the aforementioned provide solutions to analytical problems and methods for evaluating and contrasting controller performance.A controller that provides a specific performance in terms of the structured singular value may be calculated [12,13].This is the so-called (D, G-K) iteration [9], in which finding a µ-optimal controller K s such that µ(Φ u (F(jω)), K s (jω)) ≤ β, ∀ω, is transformed into the problem of finding transfer function matrices D(ω) ∀∆ and G(ω) Γ, such that, Unfortunately, even discovering local maxima is not guaranteed by this approach; however, a technique known as D-K iteration is available for complex perturbations (also implemented in MATLAB) [12,13,16].It combines H inf synthesis and µ-analysis and often produces positive results.An upper limit on µ in terms of the scaled single value serves as the starting point, It is aimed to determine the controller, which lowers the peak over frequency of its upper limit, by alternating between minimizing DN(K s )D −1 ∞ with respect to either K s or D (while maintaining the other constant) [9].

Results and Discussion
Through the relation, the function f m (t) was produced from the wind velocity data.
where V = velocity, ρ = density, and C u = 1.2.On one side of the structure, every node is subjected to periodic sinusoidal loading pressure that simulates a severe wind.
The boundaries on the values in the frequency domain are displayed in Figure 6.This results in a deviation of the mass and stiffness matrices M, and K of about 90% from their nominal values.
As can be seen, the system is still stable and performs robustly, because, for all relevant frequencies, the upper bounds of both values remain below 1.
Additionally, we regulate the structure in the state space domain by varying the nominal values of the matrices A and B, stiffness matrix K, and mass matrix M(rel.4).Account factors are considered, such as nonlinearities and system dynamics that the modeling procedure neglects, an insufficient understanding of disturbances, the disturbances caused by the environment's effect, and the decreased accuracy of system sensor data.
On one side of the structure, every node is subjected to periodic sinusoidal loading pressure that simulates a severe wind.
The boundaries on the values in the frequency domain are displayed in Figure 6.This results in a deviation of the mass and stiffness matrices M, and K of about 90% from their nominal values.
As can be seen, the system is still stable and performs robustly, because, for all relevant frequencies, the upper bounds of both values remain below 1.Additionally, we regulate the structure in the state space domain by varying the nominal values of the matrices A and B, stiffness matrix K, and mass matrix M(rel.4).Account factors are considered, such as nonlinearities and system dynamics that the modeling procedure neglects, an insufficient understanding of disturbances, the disturbances caused by the environment's effect, and the decreased accuracy of system sensor data.
The results, as shown in Figure 7, are excellent: oscillations were suppressed even for varying prices of the system's primary matrices A and B; additionally, the oscillations were reduced by differentiating the costs of the mass and stiffness matrices (12) and preserving the piezoelectric components' voltages within their endurance ranges.Figures 7b  and 8b show the displacement of the free end of the smart structure when applying Hinfinity control (close loop with PZTvoltages)in the schematic with the blue line.The smart piezoelectric structure almost has no vibrations, and it maintains equilibrium even when the key system matrices (A, B, M, and K) have different prices.In Figure 7b with green, red, light blue and petrol line we can see the displacement of the free end of the beam with different prices of matrices A and B of our system for the open loop that means without PZT voltages.Also in Figure 8b with green, red, light blue and petrol line we can see the displacement of the free end of the beam with different prices of matrices M and K of our system for the open loop that means without PZT voltages.Figures 7 and 8 in the last graph show the changes when the PZT material properties change.The smart piezoelectric structure almost has no vibrations, and it maintains equilibrium even when the key system matrices (A, B, M, and K) have different prices.The initial parameters are the mass, the damping, and the stiffness matrix.In Figures 7 and 8 these parameters change for the open and the closed loop-this means without PZT material and with PZT material.This The results, as shown in Figure 7, are excellent: oscillations were suppressed even for varying prices of the system's primary matrices A and B; additionally, the oscillations were reduced by differentiating the costs of the mass and stiffness matrices (12) and preserving the piezoelectric components' voltages within their endurance ranges.Figures 7 and 8 show the displacement of the free end of the smart structure when applying H infinity control (close loop with PZTvoltages)in the schematic with the blue line.The smart piezoelectric structure almost has no vibrations, and it maintains equilibrium even when the key system matrices (A, B, M, and K) have different prices.In Figure 7 with green, red, light blue and petrol line we can see the displacement of the free end of the beam with different prices of matrices A and B of our system for the open loop that means without PZT voltages.Also in Figure 8 with green, red, light blue and petrol line we can see the displacement of the free end of the beam with different prices of matrices M and K of our system for the open loop that means without PZT voltages.Figures 7 and 8 in the last graph show the changes when the PZT material properties change.The smart piezoelectric structure almost has no vibrations, and it maintains equilibrium even when the key system matrices (A, B, M, and K) have different prices.The initial parameters are the mass, the damping, and the stiffness matrix.In Figures 7 and 8 these parameters change for the open and the closed loop-this means without PZT material and with PZT material.This work focuses on a specific PZT material with its properties shown in Table 1. Figure 7 (last graph) shows the changes when the PZT material properties change.
The discovered H infinity controller is 24 in order.Numerous scientists have proposed algorithms for order reduction as a result of the fact that the order of the controller, which is equal to the order of the system, is substantially higher than the order of conventional controllers such as PI and LQR.The following process will use the most widely used of these algorithms, known as Hifoo [21], which has been implemented in the Matlab environment.The main issue is to calculate a reduced-order n < 24 controller that preserves the performance of the H infinity criterion and the behavior of a full-order controller of the given system.As a mechanical input to this controller, 10 KN is taken at the free end of the structure.In Figure 9 we can see the beam-free end displacement with and without control, using Hifoo recovery time 0.05 sec (0.03 with H infinity ), the steady-state error of the order of 10 −5 m (10 −6 with H infinity ) maximum elevation 2.1 × 10 −4 (0.3 × 10 −4 with H infinity ) and vibration suppression at 90% (98% with H infinity ).In Figure 10, we can see the voltages within the piezoelectric limits of 30 volts.work focuses on a specific PZT material with its properties shown in Table 1. Figure 7 (last graph) shows the changes when the PZT material properties change.work focuses on a specific PZT material with its properties shown in Table 1. Figure 7 (last graph) shows the changes when the PZT material properties change.given system.As a mechanical input to this controller, 10 KN is taken at the free the structure.In Figure 9 we can see the beam-free end displacement with and w control, using Hifoo recovery time 0.05 sec (0.03 with Hinfinity), the steady-state erro order of 10 −5 m (10 −6 with Hinfinity) maximum elevation 2.1 × 10 −4 (0.3 × 10 −4 with Hinfi vibration suppression at 90% (98% with Hinfinity).In Figure 10, we can see the v within the piezoelectric limits of 30 volts.The frequency response of the weighting function and matching model is sh Figure 11.The graph of the function remains below unity so the controller archives performance for the given data.given system.As a mechanical input to this controller, 10 KN is taken at the free the structure.In Figure 9 we can see the beam-free end displacement with and control, using Hifoo recovery time 0.05 sec (0.03 with Hinfinity), the steady-state erro order of 10 −5 m (10 −6 with Hinfinity) maximum elevation 2.1 × 10 −4 (0.3 × 10 −4 with Hinf vibration suppression at 90% (98% with Hinfinity).In Figure 10, we can see the v within the piezoelectric limits of 30 volts.The frequency response of the weighting function and matching model is shown in Figure 11.The graph of the function remains below unity so the controller archives robust performance for the given data.

Table 1 .
Parameters of the smart beam.

Figure 2 .
Figure 2. Typical control block graph.The disturbance vector (mechanical force) d and noise vector n are the diagram's two inputs, and the control vector u and state vector x are the diagram's two outputs.It is expected in what follows that,

Figure 7 .
Figure 7. Results for matrices A and B with and without Hinf, controlling for sinusoidal external inputs, at various prices.

Figure 8 .
Figure 8. Results for matrices M (mass) and K (stiffness) at various costs, both with and without Hinf control of the external sinusoidal inputs.

Figure 7 .
Figure 7. Results for matrices A and B with and without H inf , controlling for sinusoidal external inputs, at various prices.

Figure 7 .
Figure 7. Results for matrices A and B with and without Hinf, controlling for sinusoidal external inputs, at various prices.

Figure 8 .
Figure 8. Results for matrices M (mass) and K (stiffness) at various costs, both with and without Hinf control of the external sinusoidal inputs.Figure 8. Results for matrices M (mass) and K (stiffness) at various costs, both with and without H inf control of the external sinusoidal inputs.

Figure 8 .
Figure 8. Results for matrices M (mass) and K (stiffness) at various costs, both with and without Hinf control of the external sinusoidal inputs.Figure 8. Results for matrices M (mass) and K (stiffness) at various costs, both with and without H inf control of the external sinusoidal inputs.

Figure 9 .
Figure 9. Displacement of the free end of the structure with and without Hifoo.

Figure 10 .
Figure 10.Piezoelectric voltages with reduced order controller for the first four nodes.

Figure 9 .
Figure 9. Displacement of the free end of the structure with and without Hifoo.

Figure 9 .
Figure 9. Displacement of the free end of the structure with and without Hifoo.

Figure 10 .
Figure 10.Piezoelectric voltages with reduced order controller for the first four nodes.

Figure 10 .
Figure 10.Piezoelectric voltages with reduced order controller for the first four nodes.