bibliography: => #+BIBLIOGRAPHY: here

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parent a79f30c4d8
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--- a/content/book/du19_multi_actuat_system_contr.md
+++ b/content/book/du19_multi_actuat_system_contr.md
@@ -9,7 +9,7 @@ Tags


 Reference
-: ([Du and Pang 2019](#orgfa82dec))
+: ([Du and Pang 2019](#orge6fd258))

 Author(s)
 : Du, C., & Pang, C. K.
@@ -41,11 +41,11 @@ When high bandwidth, high position accuracy and long stroke are required simulta
 Popular choices for coarse actuator are:

 -   DC motor
-   Voice coil motor (VCM)
+-   [Voice Coil Motors]({{< relref "voice_coil_actuators" >}}) (VCM)
 -   Permanent magnet stepper motor
 -   Permanent magnet linear synchronous motor

-As fine actuators, most of the time piezoelectric actuator are used.
+As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}) are used.

 In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.

@@ -81,9 +81,9 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo

 #### Secondary Actuators {#secondary-actuators}

-We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org2501c9c)) and the microactuator.
+We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org5b375a0)) and the microactuator.

-<a id="org2501c9c"></a>
+<a id="org5b375a0"></a>

 {{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}

@@ -105,9 +105,9 @@ There characteristics are shown on table [1](#table--tab:microactuator).

 ### Single-Stage Actuation Systems {#single-stage-actuation-systems}

-A typical closed-loop control system is shown on figure [2](#org3f9b6d4), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
+A typical closed-loop control system is shown on figure [2](#orgeb3a161), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.

-<a id="org3f9b6d4"></a>
+<a id="orgeb3a161"></a>

 {{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}

@@ -117,7 +117,7 @@ A typical closed-loop control system is shown on figure [2](#org3f9b6d4), where
 Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
 The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.

-<a id="org59cb446"></a>
+<a id="org1f9ca75"></a>

 {{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}

@@ -143,7 +143,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a

 ### Control Schemes {#control-schemes}

-A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org371fae9).
+A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#orgd6f5782).

 -   \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
 -   \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@@ -151,7 +151,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
 -   \\(n\\) is the measurement noise
 -   \\(d\_u\\) stands for external vibration

-<a id="org371fae9"></a>
+<a id="orgd6f5782"></a>

 {{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}

@@ -173,14 +173,14 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
 And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
 Thus, the dual-stage system control design can be decoupled into two independent controller designs.

-Another type of control scheme is the **parallel structure** as shown in figure [5](#org3d4cd09).
+Another type of control scheme is the **parallel structure** as shown in figure [5](#org3fd07ea).
 The open-loop transfer function from \\(pes\\) to \\(y\\) is
 \\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]

 The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
 \\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]

-<a id="org3d4cd09"></a>
+<a id="org3fd07ea"></a>

 {{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}

@@ -190,7 +190,7 @@ Because of the limited displacement range of the secondary actuator, the control
 ### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}

 \\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
-The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org6dcd465) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
+The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org7277927) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.

 For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and

@@ -200,11 +200,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha

 is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).

-<a id="org6dcd465"></a>
+<a id="org7277927"></a>

 {{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}

-Equation [1](#orgd1210d7) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
+Equation [1](#org2fdb8dc) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
 One form of \\(W(s)\\) is taken as

 \begin{equation}
@@ -217,16 +217,16 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap

 The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
 But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
-As seen in figure [7](#orgbe7f7d1), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
+As seen in figure [7](#org038c9b4), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.

-<a id="orgbe7f7d1"></a>
+<a id="org038c9b4"></a>

 {{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}

-The sensitivity functions are shown in figure [8](#orgfed486e), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
+The sensitivity functions are shown in figure [8](#orged3fc33), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
 This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.

-<a id="orgfed486e"></a>
+<a id="orged3fc33"></a>

 {{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}

@@ -259,13 +259,13 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P

 ### Control Strategy and Controller Design {#control-strategy-and-controller-design}

-Figure [9](#orga4074a3) shows the control structure for the three-stage actuation system.
+Figure [9](#orgd7a95ee) shows the control structure for the three-stage actuation system.

 The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
 The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
 The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.

-<a id="orga4074a3"></a>
+<a id="orgd7a95ee"></a>

 {{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}

@@ -294,15 +294,15 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
 The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.

 The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
-The open-loop frequency responses of the three stages are shown on figure [10](#org11a6581).
+The open-loop frequency responses of the three stages are shown on figure [10](#org50938c4).

-<a id="org11a6581"></a>
+<a id="org50938c4"></a>

 {{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}

-The obtained sensitivity function is shown on figure [11](#org58e9561).
+The obtained sensitivity function is shown on figure [11](#org47b7e75).

-<a id="org58e9561"></a>
+<a id="org47b7e75"></a>

 {{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}

@@ -317,7 +317,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
 Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
 Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.

-For the three-stage control architecture as shown on figure [9](#orga4074a3), the position error is
+For the three-stage control architecture as shown on figure [9](#orgd7a95ee), the position error is
 \\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
 The control signals and positions of the actuators are given by

@@ -333,11 +333,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.

 ### Different Configurations of the Control System {#different-configurations-of-the-control-system}

-A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org066c259)).
+A decoupled control structure can be used for the three-stage actuation system (see figure [12](#orgc297a81)).

 The overall sensitivity function is
 \\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
-with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org7c7e2b1) and
+with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org77f79f5) and
 \\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]

 Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@@ -346,7 +346,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
 The open-loop transfer function of the overall system is
 \\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]

-<a id="org066c259"></a>
+<a id="orgc297a81"></a>

 {{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}

@@ -358,9 +358,9 @@ The control signals and the positions of the three actuators are
  u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
 \end{align\*}

-The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org8c9fc90)).
+The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org10ab429)).

-<a id="org8c9fc90"></a>
+<a id="org10ab429"></a>

 {{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}

@@ -669,6 +669,7 @@ Using PZT elements as a sensor to deal with high-frequency vibration beyond the
 As a more advanced concept, PZT elements being used as actuator and sensor simultaneously has also been addressed in this book with detailed scheme and controller design methodology for effective utilization.


+
 ## Bibliography {#bibliography}

-<a id="orgfa82dec"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
+<a id="orge6fd258"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.