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Abstract

Vibration isolation systems based on hybrid electromagnets, consisting of electromagnet and permanent magnet, have a potential usage in many industrial areas, such as clean room design, transportation, semiconductor manufacturing, suspension systems, and robotic surgery due to providing mechanical contact free vibration isolation. Using permanent magnets in the electromagnet structure has some crucial advantages, such as a minimized volume and a more compact structure. Furthermore, the essential force for levitation of vibration isolation stage can be generated by only the permanent magnet(s), which means, by using hybrid electromagnets, magnetic levitation can be achieved with considerably low energy consumption against possible vibrations. This property is called zero-power behavior. However, the main problems of magnetic levitation process are as follows: it has highly nonlinear nature even if it can be linearized; it has unstable pole(s), which makes the system vulnerable in terms of stability. In recent years, linear matrix inequality-based design of controllers has received considerable attention and become very popular due to their ability to satisfy multiobjective design requirements. However, an observer-based H₂ controller design for a vibration isolation system having hybrid electromagnets has not been considered yet. Therefore, the linear matrix inequality-based controller is employed to minimize the effect of disturbances on the following objectives, such as vibration isolation, zero-power property, and protection of the levitation gap. The effectiveness of the proposed method is shown with the numerical simulation studies and compared with classical Linear Quadratic Regulator (LQR) approach.

Keywords

Linear matrix inequalities active vibration control observer-based control magnetic levitation zero power

Introduction

In the vibration isolation perspective, there are two types of disturbances that have to be considered to design a vibration isolation system.^1–3 First type is direct disturbance—it occurs by the force that is directly applied on the vibration isolator stage, and the second type is ground-induced disturbance—it occurs by the vibrations that come from the ground. These disturbances can somehow be handled using passive suspension systems.⁴ However, at this point, many performance and sustainability problems arise. The main reason is the existence of an inevitable trade-off, since both the disturbances cannot be rejected at the same time by a simple passive suspension system. If a suspension design is stiff enough to handle direct disturbances, ground disturbance isolation performance of the system decreases. If a suspension design is soft enough to suppress ground disturbances, it is not able to cope with direct disturbances. In the light of above discussions, an active suspension system is needed to achieve satisfactory disturbance attenuation performance.^5–7

Most of active suspension systems are required to use high cost actuators and transducers.^8,9 Moreover, they are not capable with low energy consumption requirements.^4,10 At this point, vibration isolation systems based on hybrid electromagnets have become a good alternative,^11–16 because in hybrid electromagnet structure, permanent magnets behave as if they were additional current-source equivalents. Yet, electromagnet and permanent magnet both are highly nonlinear in nature,¹⁷ which means that the whole system may show undesired performance under varying disturbances. The main reason is that disturbances enforce the levitation gap which moves away from its linearization or operating point. Unless the levitation gap is ensured, the system tends to fall down or stick up. Consequently, the performance requirements, such as vibration isolation and zero-power property, cannot be satisfied as well. Therefore, in this study, an observer-based H₂ controller is used to deal with multiobjective performance requirements in the presence of disturbances varying at different frequencies.

In the last few decades, linear matrix inequalities (LMIs) have been intensively used as a strong tool in the field of control theory. Many problems, such as H₂ and $H_{\infty}$ state feedback and dynamic output feedback controller synthesis,¹⁸ analysis and design of robustness against parametric uncertainties,^19,20 and stability of time delay systems^21,22 can all be reduced to convex or quasi-convex multiobjective problems that contain LMIs.²³ Due to successful improvements of efficient and reliable algorithms^24,25 for solving LMIs, such as interior point method,²⁶ these problems can now be solved in polynomial time.

LMIs have recently been used to address few problems in magnetic levitation, such as disturbance rejection for position control of a simple levitated mass^27,28 and gain-scheduling control for a magnetically levitated rotor.²⁹ However, a detailed study for a vibration isolation system that can both suppress direct and ground disturbances in LMI framework has not been considered so far. In this study, for the controller design, it is aimed to meet multiobjective requirements which can be summarized as follows:

The levitation gap has to be protected.

Position of the vibration isolation stage must be least affected by disturbances.

The zero-power property has to be provided.

The rest of the study is organized as follows: first, the vibration isolator mechanism dynamics, consisting of hybrid electromagnet dynamics, isolator dynamics, and state-space representation, are introduced. Then, an observer-based H₂ controller design with LMIs is given. Finally, the numerical simulation studies are given with the discussions.

Mathematical modeling of vibration isolator mechanism

In this section, state-space representation of the overall system is derived to enable controller design with LMIs. First, dynamics of the hybrid electromagnets are presented. Then, mechanical dynamics of the overall isolator stage are written down by Newton laws. Finally, control-oriented state-space representation which is tailored to design an observer-based H₂ controller is given.

Hybrid electromagnet dynamics

In the analysis of a single coil, magnetic resistance, hysteresis of the iron core, eddy currents, flux leakage, and fringing effects are assumed to be negligible. So that, the electromagnetic force for vertical direction is obtained as follows

F_{e} (x_{g}, i) = ϵ {(\frac{i + I_{m}}{x_{g} + ℓ_{m} / μ_{r}})}^{2}

(1)

where ϵ is the configuration parameter of the electromagnet, i is the coil current,

x_{g} = x_{2} - x_{1}

is the equivalent air gap between the two masses,

ℓ_{m}

is the length of permanent magnets, μ_r is the relative permeability of the permanent magnet, and I_m is the equivalent current representation of the permanent magnet.

In Figure 1, highly nonlinear behavior of equation (1) is shown for the system parameters $I_{m} = 13.44$ A, $ℓ_{m} = 0.003$ m, $μ_{r} = 1$ H, and $ϵ = 15.629 \times 10^{- 6}$ N²/A².

Figure 1.

Nonlinear behavior of electromagnetic attraction force on vertical motion.

The levitation gap used in this study is 5 mm, and the zero-power property occurs around 0 A. Therefore, the linearization approach is applied to equation (1) around $x_{g 0} = 0.005$ m and $i_{0} = 0$ A as follows

K_{x_{g}} (x_{g}, i) = \frac{\partial F_{e}}{\partial x_{g}} = - 2 ϵ \frac{{(i + L_{m})}^{2}}{{(x_{g} + ℓ_{m} / μ_{r})}^{3}}

(2)

K_{i} (x_{g}, i) = \frac{\partial F_{e}}{\partial i} = 2 ϵ \frac{(i + L_{m})}{{(x_{g} + ℓ_{m} / μ_{r})}^{2}}

(3)

where

K_{x_{g}}

is the gap factor and K_i is the current factor. Then, equation (1) can be approximated as

F_{e} (x_{g}, i) ≅ K_{x_{g}} (x_{g 0}, i_{0}) Δ x_{g} + K_{i} (x_{g 0}, i_{0}) Δ i + F_{e} (x_{g 0}, i_{0})

(4)

where

x_{g 0}

and i₀ are the levitation gap and the coil current values at the linearization point, respectively.

Δ x_{g} = x_{g} - x_{g 0}

is the difference between the actual levitation gap and the linearization point. According to Newton’s second law, the rate of change of momentum of the levitated mass on the vertical axis can be written as follows

m_{2} \frac{d^{2} x_{2}}{d t^{2}} = - F_{e} (x_{g}, i) + m g + F_{d}

(5)

Here, m₂ is mass of the levitated object, g is the gravitational acceleration constant, and F_d is the disturbance.

Electrical dynamics of the system is given below

\frac{d i}{d t} = \frac{K_{V}}{L} \frac{d x_{g}}{d t} - \frac{R}{L} i + \frac{1}{L} V

(6)

Here, $K_{V} / L$ is a function of the gap factor and current factor as follows: $K_{V} / L = K_{x_{g}} (x_{g}, i) / K_{i} (x_{g}, i)$ .

Isolator dynamics

The two mass vibration isolation system combined with a hybrid electromagnet in Figure 2 was first proposed by Mizuno et al.¹³ Here, x₀ is the ground disturbance displacement, x₁ is the position of middle mass and x₂ is the position of the isolation stage, and x_g is the levitation gap

m_{1} \frac{d^{2} Δ x_{1}}{d t^{2}} + c (\frac{d Δ x_{1}}{d t} - \frac{d Δ x_{0}}{d t}) + k (Δ x_{1} - Δ x_{0}) = K_{x_{g} 0} Δ x_{g} + K_{i 0} Δ i

(7)

Figure 2.

Two mass vibration isolation system.

Here, $Δ x_{1}$ and $Δ x_{2}$ are the relative displacements from linearization point, $Δ x_{g} = Δ x_{2} - Δ x_{1}$ . $K_{x g 0} = K_{x_{g}} (x_{g 0}, i_{0})$ is the gap constant and $K_{i 0} = K_{i} (x_{g 0}, i_{0})$ is the current constant. Applying the linearization on equation (5) around an equilibrium point

m_{2} \frac{d^{2} Δ x_{2}}{d t^{2}} = - K_{x_{g} 0} Δ x_{g} - K_{i 0} Δ i + F_{d}

(8)

Here, m₂ is mass of the levitated object, g is the gravitational acceleration constant, and F_d is the disturbance; $K_{x g 0} = K_{x_{g}} (x_{g 0}, i_{0})$ is the gap constant and $K_{i 0} = K_{i} (x_{g 0}, i_{0})$ is the current constant.

Electrical dynamics of the system is given below

\frac{d Δ i}{d t} = \frac{K_{x_{g 0}}}{K_{i 0}} \frac{d Δ x_{g}}{d t} - \frac{R}{L} Δ i + \frac{1}{L} Δ V

(9)

Control-oriented state-space model

Any linear time invariant system can be represented by a state-space framework as given below

\frac{d x}{d t} = A x + B_{1} u + B_{2} w

(10)

Here, $A \in ℝ^{n \times n}$ is the state matrix, $B_{1} \in ℝ^{n \times r}$ is the control input matrix, and $B_{2} \in ℝ^{n \times p}$ is the exogenous input matrix which is used to describe the effect of external signals such as disturbance and reference trajectory. Higher order terms in equations (7) to (9) are left alone on the left-hand side as follows

\begin{matrix} \frac{d^{2} Δ x_{1}}{d t^{2}} = - \frac{c}{m_{1}} \frac{d Δ x_{1}}{d t} + \frac{c}{m_{1}} \frac{d Δ x_{0}}{d t} - \frac{k}{m_{1}} Δ x_{1} + \frac{k}{m_{1}} Δ x_{0} \\ + \frac{K_{x g 0}}{m_{1}} Δ x_{2} - \frac{K_{x g 0}}{m_{1}} Δ x_{1} + \frac{K_{i 0}}{m_{1}} Δ i \end{matrix}

(11)

\frac{d^{2} Δ x_{2}}{d t^{2}} = - \frac{K_{x g 0}}{m_{2}} Δ x_{2} + \frac{K_{x g 0}}{m_{2}} Δ x_{1} - \frac{K_{i 0}}{m_{2}} Δ i + \frac{1}{m_{2}} F_{d}

(12)

\frac{d Δ i}{d t} = \frac{K_{x g 0}}{K_{i 0}} \frac{d Δ x_{2}}{d t} - \frac{K_{x g 0}}{K_{i 0}} \frac{d Δ x_{1}}{d t} - \frac{R}{L} Δ i + \frac{1}{L} Δ V

(13)

In order to rewrite the vibration isolator dynamics in state-space representation, state vector $x \in ℝ^{n}$ must be determined. As mentioned before, the position of mass-1 and mass-2 is desired to be asymptotically stable, the levitation gap expected to have reference tracking capability, and the system has to show zero-power property. Therefore, the state vector is chosen as

x = {[\begin{matrix} Δ x_{1} & Δ x_{2} & \frac{d Δ x_{1}}{d t} & \frac{d Δ x_{2}}{d t} & \int (r - Δ x_{g}) d t & Δ i \end{matrix}]}^{T}

(14)

Here, r is the reference trajectory which is desired to be tracked by $Δ x_{g}$ . Then, the control input vector $u \in ℝ^{r}$ and the exogenous input vector $w \in ℝ^{p}$ are given as

u = Δ V, w = {[\begin{matrix} Δ x_{0} & \frac{Δ x_{0}}{d t} & F_{d} & r \end{matrix}]}^{T}

(15)

By the use of equations (10) to (15), the state-space matrices are in the form of

A = [\begin{matrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ - \frac{k + K_{x g 0}}{m_{1}} & \frac{K_{x g 0}}{m_{1}} & - \frac{c}{m_{1}} & 0 & 0 & \frac{K_{i 0}}{m_{1}} \\ \frac{K_{x g 0}}{m_{2}} & - \frac{K_{x g 0}}{m_{2}} & 0 & 0 & 0 & - \frac{K_{i 0}}{m_{2}} \\ 1 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{K_{x g 0}}{K_{i 0}} & \frac{K_{x g 0}}{K_{i 0}} & 0 & - \frac{R}{L} \end{matrix}]

(16)

B_{1} = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & \frac{1}{L} \end{matrix}]}^{T}

(17)

B_{2} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{k}{m_{1}} & \frac{c}{m_{1}} & 0 & 0 \\ 0 & 0 & \frac{1}{m_{2}} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}]

(18)

In order to design an observer-based controller, measured outputs vector has to be defined according to practically available sensors and transducers. Measured output vector is the linear function of state and input variables as follows

y = C_{1} x + D_{11} u + D_{12} w

(19)

Here, $C_{1} \in ℝ^{h \times n}, D_{11} \in ℝ^{h \times r}$ , and $D_{12} \in ℝ^{h \times p}$ are measured output matrices with appropriate dimensions. Current sensors are popularly used in mechatronics applications due to their low-cost property. Air gap sensors, especially capacitive ones, are very appropriate for maglev process. And velocity parameter can be obtained by numerical integration methods from accelerometer outputs by using many methods. Therefore, the measured output vector y is given as follows

y = {[\begin{matrix} Δ x_{1} - Δ x_{2} & \frac{d Δ x_{1}}{d t} & \frac{d Δ x_{2}}{d t} & \int r - (Δ x_{2} - Δ x_{1}) & Δ i \end{matrix}]}^{T}

(20)

Then to construct a measured output (20), the following matrices are needed to be selected as

C_{1} = [\begin{matrix} - 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

(21)

D_{11} = 0_{5 \times 1}, D_{12} = 0_{5 \times 4}

(22)

In the controller design, effect of the exogenous inputs on some certain output variables is required to be minimized. The certain output variables are named as controlled outputs and shown as $z \in ℝ^{m}$

z = C_{2} x + D_{21} u + D_{22} w

(23)

Here, $C_{2} \in ℝ^{m \times n}, D_{21} \in ℝ^{m \times r}$ , and $D_{22} \in ℝ^{m \times p}$ are controlled output matrices with appropriate dimensions. Controlled outputs vector is supposed to be chosen carefully to satisfy performance requirements. First, it is desired that mass-2 must be unaffected by disturbances to ensure vibration isolation property. Second, the levitation gap has to be protected to stay within linearized operating point. Finally, produced control signal and current variable need to be minimized to achieve zero-power property in the absence of disturbances. Therefore, the controlled output vector is written down as follows

z = {[\begin{matrix} Δ x_{2} & \int r - (Δ x_{2} - Δ x_{1}) & Δ V \end{matrix}]}^{T}

(24)

Then to construct a controlled outputs vector in the form of equation (24), the controlled output matrices are needed to be selected as

C_{2} = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(25)

D_{21} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], D_{22} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

(26)

Finally, the physical parameters used throughout the study are given in Table 1.

Table 1.

Physical parameter values.

Parameter	Value	Unit	Parameter	Value	Unit
m ₁	4.5	$(kg)$	m ₂	4.5	$(kg)$
ϵ	$15.629 \times 10^{- 6}$	$(N^{2} A^{2})$	I_m	13.44	$(A)$
R	1.6	$(Ω)$	L	0.016	$(H)$
$K_{x g 0}$	$- 11, 028$	$(N/m)$	$K_{i 0}$	6.56	$(N/A)$
z ₀	5	$(mm)$	i ₀	0	$(A)$
$ℓ_{m}$	3	$(mm)$	μ_r	1	$(H)$
k	1500	$(N/m)$	c	65.727	$(N s/m)$

Observer-based H₂ controller design

In this paper, to provide an active vibration isolation, an observer-based H₂ controller design via LMIs framework has been employed. Based on the overall system model, an H₂ full-state feedback controller could be designed alone. However, all the state variables belonging to the vibration isolator are not available for feedback. Despite the fact that noncontact electromagnetic and/or laser-based transducers can be used to sense relative position in the presence of fixed reference frame, absolute positions are not accessible due to absence of fixed reference frame. Therefore, an observer-based control strategy which uses estimated states rather than actual ones has been used to achieve optimal vibration isolation in terms of H₂ norm performance index.

Problem formulation

Consider an open-loop state-space system given by equation (27)

\begin{array}{l} \frac{d x}{d t} = A x + B_{1} u + B_{2} w \\ y = C_{1} x + D_{11} u + D_{12} w \\ z = C_{2} x + D_{21} u + D_{22} w \end{array}

(27)

An observer-based control law can be written as equation (28)

\begin{array}{l} \frac{d \hat{x}}{d t} = A \hat{x} + B_{1} u - L (y - \hat{y}) \\ \hat{y} = C_{1} \hat{x} + D_{11} u \\ u = K \hat{x} \end{array}

(28)

Here, $L \in ℝ^{n \times h}$ is the observer gain and $K \in ℝ^{r \times n}$ is the state feedback controller gain. In the observer-based control law (28), estimated states rather than actual ones feed to the controller gain. In order to achieve high performance, estimated states must be received instantaneously by the controller. In addition, the difference between the estimated and actual states has to be converged to zero. Therefore, error dynamics can be formulated as

\begin{array}{l} \frac{d e}{d t} = (A + L C_{1}) e + (B_{2} + L D_{12}) w \\ z_{e} = C_{3} e \end{array}

(29)

Here, $e = x - \hat{x}$ is the error variable and $z_{e} \in ℝ^{n}$ is the vector to choose which observation error must be less affected by disturbances. By substituting the open-loop system equation (27) into the control law (28), the closed-loop system is obtained as follows

\begin{array}{l} [\begin{matrix} \frac{d x}{d t} \\ \frac{d e}{d t} \end{matrix}] = [\begin{matrix} A + B_{1} K & - B_{1} K \\ 0 & A + L C_{1} \end{matrix}] [\begin{matrix} x \\ e \end{matrix}] + [\begin{matrix} B_{2} \\ B_{2} + L D_{12} \end{matrix}] [w] \\ [\begin{matrix} z \\ z_{e} \end{matrix}] = [\begin{matrix} C_{2} + D_{21} K & - D_{21} K \\ 0 & C_{3} \end{matrix}] [\begin{matrix} x \\ e \end{matrix}] + [\begin{matrix} D_{22} \\ 0 \end{matrix}] [w] \end{array}

(30)

The main objective is designing an observer-based control law, such that following conditions hold:

The closed-loop system (30) is stable which ensures the stability of the open-loop system (27) and the error dynamics equation (29) simultaneously.

In order to ensure a finite value of H₂ norm, D₂₂ is $0_{m x r}$ .

H₂ state feedback controller design

In this section, first it is assumed that all the state variables are available for feedback. Thus, the state feedback control law is given by equation (31)

u = K x

(31)

Then, by substituting the state feedback control law (31) into the open-loop system (27) under the assumption of full-state feedback, the resulted closed-loop system

\begin{array}{l} \frac{d x}{d t} = (A + B_{1} K) x + B_{2} w \\ z = (C_{2} + D_{21} K) x + D_{22} w \end{array}

(32)

is derived. The main requirements can be stated as follows:

The closed-loop system (32) is asymptotically stable which indicates that all eigenvalues of $A + B_{1} K$ have negative real parts.

Vibration isolation performance is maximized by minimizing the H₂ norm between exogenous input w and the controlled outputs z.

Closed-loop poles are located to the left side of $- α_{c}$ to ensure speed of transient responses.

Closed-loop poles are located to the right side of $- β_{c}$ to avoid excessive control signals magnitudes and chattering-like characteristics.

Then, the following theorem presents a full-state feedback H₂ controller design.

Theorem 1³⁰: For a given positive scalar γ, α_c, and β_c, the closed-loop system (32) is asymptotically stable with H₂ norm less than γ and the closed-loop system poles are shifted to the left side of $- α_{c}$ and the right side of $- β_{c}$ , if there exist positive definite matrix $Y \in ℝ^{n \times n}$ and $Z \in ℝ^{h \times h}$ ; and a rectangular matrix $V \in ℝ^{r \times n}$ subject to following convex optimization problem

minγ; s.t.equations (33) to (37)

A Y + B_{1} V + {(A Y + B_{1} V)}^{T} + B_{2} + B_{2}^{T} < 0

(33)

[\begin{matrix} - Z & - C_{2} Y + D_{21} V \\ {(C_{2} Y + D_{21} V)}^{T} & - Y \end{matrix}] < 0

(34)

T r a c e (Z) < γ^{2}

(35)

A Y + B_{1} V + {(A Y + B_{1} V)}^{T} + 2 α_{c} Y < 0

(36)

A Y + B_{1} V + {(A Y + B_{1} V)}^{T} + 2 β_{c} Y > 0

(37)

Then, the control law $u = K x = V Y^{- 1}$ is the state feedback controller having pole location constraints.

H₂ observer design

In this section, LMI-based design of a full-order observer is presented. For an H₂ observer design problem, the main requirements can be stated as follows:

Making the observation error dynamics (29) asymptotically stable which means that all eigenvalues of $A + L C_{1}$ has negative real parts.

Keeping the H₂ norm between z_e and w as small as possible.

Closed-loop poles are located to the left side of $- α_{0}$ which is expected to be less than $- β_{c}$ , since the convergence of the estimated states must be faster than the fastest pole of the controlled plant.

Closed-loop poles are located to the right side of $- β_{0}$ to avoid implementation problems due to the need of very high sampling frequency.

The following theorem formulates the H₂ observer design problem.

Theorem 2³⁰: For a given positive scalar $φ$ , α₀, and β₀, the error dynamics (29) is asymptotically stable with H₂ norm less than $φ$ and poles of the error dynamics are placed to the left side of $- α_{0}$ and the right side of $- β_{0}$ , if there exist positive definite matrix $X \in ℝ^{n \times n}$ and $Q \in ℝ^{n \times n}$ ; and a rectangular matrix $W \in ℝ^{n \times m}$ subject to following convex optimization problem

minγ; s.t.equations (38) to (42)

[\begin{matrix} X A + W C_{1} + {(X A + W C_{1})}^{T} & X B_{2} + W D_{12} \\ {(X B_{2} + W D_{12})}^{T} & - I \end{matrix}] < 0

(38)

[\begin{matrix} - Q & C_{3} \\ C_{3}^{T} & - X \end{matrix}] < 0

(39)

T r a c e (Q) < φ^{2}

(40)

X A + W C_{1} + {(X A + W C_{1})}^{T} + 2 α_{0} X < 0

(41)

X A + W C_{1} + {(X A + W C_{1})}^{T} + 2 β_{0} X > 0

(42)

Then, $L = W X^{- 1}$ is the full-order observer gain having pole location constraints.

Construction of the observer-based controller

In this section, the state feedback controller and the full-order observer are designed. To compute the state feedback controller positive scalars α_c is chosen to be 5 and β_c is chosen to be 200. Then by using the YALMIP parser²⁴ and SeDuMi solver,²⁵ the H₂ full-state feedback controller gain can be computed with Theorem 1 as follows

K = [\begin{matrix} - 1.0146 \times 10^{4} & 1.1388 \times 10^{4} & - 117.0985 \dots & 236.7465 & - 1.9058 \times 10^{4} & - 2.7942 \end{matrix}]

(43)

Then, the resulting closed-loop pole placements are given by the following equation

σ (A + B_{1} K) = (- 146.16, - 84.92, - 5.30 \pm 11.89 j, - 40.81, - 6.75)

(44)

It is apparently seen that closed-loop poles are successfully located at the desired strip region in complex plane. Thereafter, to compute the full-order observer positive scalars, α₀ is chosen to be 250 and β₀ is chosen to be 1500. In addition, all the observation error variables are equally weighted by taking. Then, by using the YALMIP parser²⁴ and SeDuMi solver,²⁵ the H₂ observer gain can be computed with Theorem 2 as follows

L = [\begin{matrix} - 9.7921 \times 10^{3} & 1.4628 \times 10^{3} & - 0.8354 & - 1.1900 \times 10^{- 6} & - 1.5007 \\ - 1.0254 \times 10^{4} & 1.4625 \times 10^{3} & - 0.4617 & 1.1921 \times 10^{- 6} & - 1.5805 \\ 1.4595 \times 10^{4} & - 1.7354 \times 10^{3} & 0.1825 & - 4.9005 \times 10^{- 9} & 0.4335 \\ - 2.4514 \times 10^{3} & - 6.2166 \times 10^{- 6} & - 463.4178 & - 1.9931 \times 10^{- 7} & 3.2788 \times 10^{- 7} \\ 1 & 3.1505 \times 10^{- 9} & - 1.0311 \times 10^{- 5} & - 502.6131 & - 1.7321 \times 10^{- 7} \\ - 0.0068 & - 1.6800 \times 10^{3} & 1.6815 \times 10^{3} & - 5.9347 \times 10^{- 6} & - 361.7637 \end{matrix}]

(45)

Then, the resulting closed-loop pole placements are given by the following equation

σ (A + L C_{1}) = (- 1383.9, - 339.6, - 462.6 \pm 1.2 j, - 488.2, - 502.6)

(46)

It is apparently seen that closed-loop poles are successfully located at the desired strip region in complex plane. The eigenvalues of the state feedback controller and observer can be seen in Figure 3.

Figure 3.

The observer eigenvalues and the controller eigenvalues on different D stability regions.

Note that stability of the closed-loop system is guaranteed since the diagonal entries of the A_cl in equation (30) are composed of $A + L C_{1}$ and $A + B_{1} K$ which are Hurwitz stable. Finally, schematic representation of the observer-based controller is depicted in Figure 4.

Figure 4.

Observer-based control scheme.

To validate and compare the performance of proposed observer-based controller with classical LQR controller, the design equations for a LQR controller can be obtained as in Theorem 3.

Theorem 3³¹: LQR controller design problem is to find an optimal full-state feedback control law that minimizes the quadratic cost function

J = \int_{0}^{\infty} (x^{T} Q x + u^{T} R u) d t

(47)

where performance weighting matrices are symmetric and positive semidefinite

Q = Q^{T} ⩾ 0

and a symmetric positive-definite matrix

R = R^{T} > 0

. Solution of algebraic Riccati equation

S A + A^{T} S + Q + S B_{2} R^{- 1} B_{2}^{T} S = 0

(48)

provides the optimal solution. Then, the LQR control law can be obtained as

u = - R^{- 1} B_{2}^{T} S x

(49)

The performance weighting matrices of the LQR controller are selected as

Q = d i a g (0, 1, 0, 0, 2 \times 10^{8}, 0); R = 1

(50)

Then, the resulting LQR controller gain is given by equation (51)

K_{2} = [\begin{matrix} - 9.8027 \times 10^{3} & 1.0408 \times 10^{4} & - 150.3808 … & 179.8427 & - 1.4142 \times 10^{4} & - 2.6397 \end{matrix}]

(51)

Simulation results and discussions

In this section, the proposed vibration isolation system is numerically tested for static and dynamic disturbances. First, vibrations of the isolation stage, air gap, and current variations under static and dynamic disturbances are analyzed in time domain. Thereafter, bode plots are given to investigate system responses under more generalized disturbance scenarios.

In the static disturbance case, direct disturbance F_d is chosen as 5 N stepwise input app. Reference value $r = Δ x_{g}$ is given as −1 mm which corresponds to $x_{g} = 4 mm$ , since equation (1) is linearized around 5 mm. Throughout the simulations, system is exposed to disturbances for 2 s along. For the proposed observer-based H₂ controller and LQR, vibration isolation performance under static disturbances in terms of position and velocity of mass-2, levitation/air gap, and current is given in Figure 5.

Figure 5.

Isolator responses under stepwise direct disturbance and levitation gap reference tracking requirements at the same time.

As can be seen from Figure 5, both controllers provide zero-power property after the disturbances disappeared. However, proposed H₂ controller has better vibration isolation performance, since less overshoot is obtained in both $Δ x_{2}$ and $d Δ x_{2} / d t$ responses. Thanks to the reference tracking capability the isolator can operate under even a stepwise direct disturbance without exceeding the allowable levitation gap limits.

In order to investigate vibration isolation performance under dynamic loading case, ground displacement x₀ is taken as $x_{0} = 5 sin (4 \times 2 π t)$ (mm), direct disturbance F_d is taken as a periodic signal with same frequency and the magnitude with 1 N. Vibration isolation performance of both the controllers is given in Figure 6.

Figure 6.

Isolator responses under ground disturbance of $5 sin (4 \times 2 π t)$ (mm) and direct disturbance of $1 sin (4 \times 2 π t)$ (N).

Figure 6 indicates that both controllers have more than 50% suppression rate on ground displacement disturbances even in the simultaneous excitation of direct disturbances. Despite the fact that both controllers provide nearly identical isolation performance, H₂ controller has significantly superior responses on levitation gap and coil current. To investigate the performance under the direct disturbances of 1–5 N interval with the $x_{0} = 5 sin (4 \times 2 π t)$ (mm), Figure 7 is given.

Figure 7.

Isolator responses under ground disturbance of $5 sin (4 \times 2 π t)$ (mm) and direct disturbance with $1 - 5 sin (4 \times 2 π t)$ (N). LQR: Linear Quadratic Regulator.

As can be seen in Figure 7, LQR controller requires much higher control effort in terms of current coil response. In Figure 8, effect of the disturbances on the isolation stage vibrations and current can be seen for the entire frequency domain. Effect of the ground disturbances and direct disturbances is considerably attenuated above 2 Hz. It is well known that realistic disturbances are usually composed of sine wave components much faster than 2 Hz. Therefore, the proposed system has quite satisfactory isolation performance with very low power consumption and safely operated levitation gap, on the frequency range of interest due to the practical considerations.

Figure 8.

Bode plots for mass-2 vibrations and levitation gap.

Remark. A full-state feedback strategy usually provides better control performance when compared with an observer-based design. In this study observer-based H₂ controller achieves practically more desirable performance in many aspects. First, better vibration damping with smaller overshoots is obtained under the stepwise disturbance excitations. Second, although very similar suppression rates are achieved for periodic excitations, more satisfactory energy consumption levels are obtained in terms of current coil responses. In the light of above-mentioned discussions, observer-based H₂ controller is a more suitable choice to design a vibration isolator operating under disturbances.

Robust stability analysis

In this subsection, robust stability of the proposed observer-based H₂ controller and classical LQR controller is analyzed.

For magnetic levitation applications, levitated object’s mass is the most important parameter due to its variable value in different scenarios. Therefore, robust stability analysis is conducted for m₂ values in the range of 1.5–7.5 kg.

In Tables 2 and 3, natural frequencies, ω_n, are given for the closed-loop systems with observer-based controller and classical LQR controller. Moreover, arithmetic means and standard deviations are listed to quantitatively describe ω_n variations against different m₂ values. Here, μ denotes arithmetic mean and σ denotes standard deviation.

Table 2.

Natural frequencies, ω_n (rad/s), for the closed-loop system with observer-based controller.

	m ₂
$#$	1.5 (kg)	2.5 (kg)	3.5 (kg)	4.5 (kg)	5.5 (kg)	6.5 (kg)	7.5 (kg)	$μ (ω_{n})$	$σ (ω_{n})$
1	222.41	169.59	138.50	146.15	182.29	198.00	207.79	180.68	31.29
2	222.41	169.59	138.50	84.92	50.73	47.31	45.18	108.38	69.82
3	16.27	23.52	29.49	40.80	50.73	47.31	45.18	36.19	13.16
4	16.27	15.01	13.92	6.74	6.70	6.67	6.64	10.28	4.52
5	19.25	15.01	13.92	13.02	12.26	11.63	11.10	13.74	2.77
6	6.88	6.83	6.78	13.02	12.26	11.63	11.10	9.79	2.82
7	1400.25	1400.26	1400.26	1400.26	1400.27	1400.27	1400.27	1400.26	0.01
8	347.70	347.71	347.71	347.72	347.72	347.72	347.72	347.71	0.01
9	696.83	582.61	600.43	625.97	627.19	627.44	627.62	626.87	35.48
10	556.89	671.09	653.26	626.97	627.19	627.44	627.62	627.07	35.47
11	625.76	625.75	625.75	627.50	625.77	625.76	625.76	626.01	0.65
12	630.56	630.56	630.56	630.56	630.56	630.56	630.56	630.56	0.00
								$\bar{μ} (ω_{n})$	$\bar{σ} (ω_{n})$
								384.79	16.34

Table 3.

Natural frequencies, ω_n (rad/s), for the closed-loop system with classical LQR controller.

	m ₂
$#$	1.5 (kg)	2.5 (kg)	3.5 (kg)	4.5 (kg)	5.5 (kg)	6.5 (kg)	7.5 (kg)	$μ (ω_{n})$	$σ (ω_{n})$
1	191.69	150.37	126.97	146.15	171.49	183.65	191.32	165.95	25.18
2	191.69	150.37	126.97	80.14	50.26	47.09	45.06	98.80	58.23
3	25.10	28.65	33.09	40.27	50.26	47.09	45.06	38.50	9.69
4	16.07	14.88	13.89	5.34	5.33	5.31	5.30	9.45	5.18
5	16.07	14.88	13.89	13.05	12.33	11.72	11.18	13.30	1.75
6	5.40	5.38	5.36	13.05	12.33	11.72	11.18	9.20	3.61
								$\bar{μ} (ω_{n})$	$\bar{σ} (ω_{n})$
								55.87	17.27

As expected, the closed-loop system with observer-based controller is 12th-order system, while the closed-loop system with classical LQR controller is sixth-order system. Therefore, the closed-loop system with observer-based controller has 12 natural frequencies and 12 damping ratios, while the closed-loop system with classical LQR controller has 6 natural frequencies and 6 damping ratios.

As can be observed from Tables 2 and 3, average standard deviation of natural frequencies, $\bar{σ} (ω_{n})$ , is slightly lower for observer-based controller, which indicates that the proposed controller is less sensitive against the change of m₂.

In Tables 4 and 5, damping ratios, ζ, are given for the closed-loop systems with observer-based controller and classical LQR controller. Moreover, arithmetic means and standard deviations are listed to quantitatively describe ζ variations against different m₂ values.

Table 4.

Damping ratio, ζ, for the closed-loop system with observer-based controller.

	m ₂
$#$	1.5 (kg)	2.5 (kg)	3.5 (kg)	4.5 (kg)	5.5 (kg)	6.5 (kg)	7.5 (kg)	$μ (ζ)$	$σ (ζ)$
1	0.56	0.72	0.87	1.00	1.00	1.00	1.00	0.87	0.17
2	0.56	0.72	0.87	1.00	0.89	0.79	0.73	0.79	0.14
3	0.42	1.00	1.00	1.00	0.89	0.79	0.73	0.83	0.21
4	0.42	0.42	0.41	1.00	1.00	1.00	1.00	0.75	0.30
5	1.00	0.42	0.41	0.40	0.39	0.38	0.36	0.48	0.22
6	1.00	1.00	1.00	0.40	0.39	0.38	0.36	0.64	0.32
7	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.00
8	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.00
9	1.00	1.00	1.00	1.00	0.99	0.99	0.99	0.99	0.01
10	1.00	1.00	1.00	1.00	0.99	0.99	0.99	0.99	0.01
11	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.00
12	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	0.00
								$\bar{μ} (ζ)$	$\bar{σ} (ζ)$
								0.86	0.11

Table 5.

Damping ratio, ζ, for the closed-loop system with classical LQR controller.

	m ₂
$#$	1.5 (kg)	2.5 (kg)	3.5 (kg)	4.5 (kg)	5.5 (kg)	6.5 (kg)	7.5 (kg)	$μ (ζ)$	$σ (ζ)$
1	0.61	0.78	0.91	1.00	1.00	1.00	1.00	0.90	0.14
2	0.61	0.78	.91	1.00	0.95	0.89	0.85	0.86	0.12
3	1.00	1.00	1.00	1.00	0.95	0.89	0.85	0.95	0.05
4	0.36	0.33	0.31	1.00	1.00	1.00	1.00	0.71	0.35
5	0.36	0.33	0.31	0.29	0.27	0.26	0.24	0.29	0.04
6	1.00	1.00	1.00	0.29	0.27	0.26	0.24	0.58	0.39
								$\bar{μ} (ζ)$	$\bar{σ} (ζ)$
								0.72	0.18

As can be observed from Tables 4 and 5, average standard deviation of damping ratios, $\bar{σ} (ζ)$ , is considerably lower (about 38%) for observer-based controller. Furthermore, $\bar{μ} (ζ)$ is 0.86 for observer-based controller, while $\bar{μ} (ζ)$ is 0.71 for classical LQR controller.

Remark. In the light of above-mentioned discussions, it is apparently seen that the proposed controller guarantees higher ζ and much less deviated ζ, ω_n for the wide range of m₂ values.

In order to investigate effect of parameter variations in frequency domain, bode plots are given in Figure 9. In terms of vibration isolation performance, the proposed controller has smaller gains for entire values of m₂. As previously mentioned by damping ratio comparisons, classical LQR controller has weaker damping property for whole m₂ values.

Figure 9.

Frequency responses under variation of m₂.

Conclusions

This paper has provided an application of a multiobjective observer-based H₂ controller on vibration isolation system having hybrid electromagnets. By the use of an LMI-based design approach, vibration isolation, zero power, and levitation gap protection have been simultaneously achieved under both ground and direct disturbances for static and dynamic loading conditions. Robust performance and robust stability of the specified controller on the proposed isolator configuration has been investigated in both time and frequency domain against a classical LQR controller. It has been revealed that the isolator having H₂ controller is capable of mitigating the vibrations caused by both ground displacement and direct disturbances on the frequency range of interest with smaller energy consumption level. Moreover, practical limitations on the levitation gap have not been violated for extensive class of disturbances.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under the Grant Number 112M210 and by Research Fund of the Yildiz Technical University under the Project Number FDK-2018-3250.

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Observer-based H 2 controller design for a vibration isolation stage having hybrid electromagnets

Abstract

Keywords

Introduction

Mathematical modeling of vibration isolator mechanism

Hybrid electromagnet dynamics

Isolator dynamics

Control-oriented state-space model

Observer-based H2 controller design

Problem formulation

H2 state feedback controller design

H2 observer design

Construction of the observer-based controller

Simulation results and discussions

Robust stability analysis

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

Observer-based H₂ controller design

H₂ state feedback controller design

H₂ observer design