Apparatus for improved general-purpose PID and non-PID controllers

US 6,847,851 B1
Filed: 07/12/2002
Issued: 01/25/2005
Est. Priority Date: 07/12/2002
Status: Expired due to Fees

First Claim

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1. A proportional, integrative, and derivative (PID) controller comprising a proportional element, an integrative element, and a derivative element coupled together and responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, wherein the proportional element has a gain element with a gain being substantially equal to $0.72 * K_{u} * ⅇ$

-1.6Ku+1.2Ku2-.001234000198⁢

*Tu-6.117274273⁢

*10-6where K_uis the ultimate gain of the plant and T_uis the ultimate period of the plant.

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Abstract

An apparatus is provided for controlling a system to achieve a specified response. In one embodiment, the apparatus is a proportional, integrative, and derivative (PID) controller having a proportional element, an integrative element, and a derivative element coupled together. The elements respond to a reference signal and generate a control signal that causes a plant to generate a plant output. The proportional element has a gain element where the gain is a function of the ultimate gain of the plant (K_u) and the ultimate period of the plant (T_u). The controllers may also be embodied in non-PID controllers that share common elements, such as the use of: (a) Astrom-Hagglund controller output as an input for a subsequent controller; (b) internal feedback; and (c) a subsequent controller that performs a subtraction operation to generate the difference between the output of the Astrom-Hagglund controller and the output of the subsequent controller.

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50 Claims

1. A proportional, integrative, and derivative (PID) controller comprising a proportional element, an integrative element, and a derivative element coupled together and responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, wherein the proportional element has a gain element with a gain being substantially equal to $0.72 * K_{u} * ⅇ$
- -1.6Ku+1.2Ku2-.001234000198⁢
  
  *Tu-6.117274273⁢
  
  *10-6where K_uis the ultimate gain of the plant and T_uis the ultimate period of the plant.
- View Dependent Claims (2, 3, 4)
- - 2. The PID controller defined in claim 1 wherein gain on the integrative element is substantially equal to $\frac{0.72 * K_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}}}{0.59 * T_{u} * e^{\frac{- 1.3}{K_{u}} + \frac{0.38}{K_{u}^{2}}}} - .06852522843250242$
    - *KuTu.
  - 3. The PID controller defined in claim 1 or 2 wherein gain on the derivative element is substantially equal to $0.108 * K_{u} * T_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}} * e^{\frac{- 1.4}{K_{u}} + \frac{0.56}{K_{u}^{2}}} -$
    - ⁢
      
      0.002664037864⁢
      
      ⁢
      
      (eTu)log⁡
      
      (1.634220701log⁢
      
      ⁢
      
      Ku).
  - 4. The PID controller defined in claims 1 or 2 wherein the reference signal is weighted by a weight substantially equal to $0.25 * e^{\frac{0.56}{K_{u}} + \frac{- 0.12}{K_{u}^{2}}} + \frac{K_{u}}{e^{K_{u}}}$

5. A proportional, integrative, and derivative (PID) controller comprising a proportional element, an integrative element, and a derivative element coupled together and responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, wherein gain on the integrative element is substantially equal to $\frac{0.72 * K_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}}}{0.59 * T_{u} * e^{\frac{- 1.3}{K_{u}} + \frac{0.38}{K_{u}^{2}}}} - .06852522843250242$
- *KuTu,where K_uis the ultimate gain of the plant and T_uis the ultimate period of the plant.
- View Dependent Claims (6, 7)
- - 6. The PID controller defined in claim 5 wherein gain on the derivative element is substantially equal to $0.108 * K_{u} * T_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}} * e^{\frac{- 1.4}{K_{u}} + \frac{0.56}{K_{u}^{2}}} -$
    - ⁢
      
      0.002664037864⁢
      
      ⁢
      
      (eTu)log⁡
      
      (1.634220701log⁢
      
      ⁢
      
      Ku).
  - 7. The PID controller defined in claims 5 or 6 wherein the reference signal is weighted by a weight substantially equal to $0.25 * e^{\frac{0.56}{K_{u}} + \frac{- 0.12}{K_{u}^{2}}} + \frac{K_{u}}{e^{K_{u}}}$

8. A proportional, integrative, and derivative (PID) controller comprising a proportional element, an integrative element, and a derivative element coupled together and responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, wherein the proportional element has a gain element with a gain being substantially equal to $0.72 * K_{u} * ⅇ$
- -1.6Ku+1.2Ku2-.001234000198⁢
  
  *Tu-6.117274273⁢
  
  *10-6where K_uis the ultimate gain of the plant and T_uis the ultimate period of the plant, and further wherein gain on the integrative element is substantially equal to $\frac{0.72 * K_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}}}{0.59 * T_{u} * e^{\frac{- 1.3}{K_{u}} + \frac{0.38}{K_{u}^{2}}}} - .06852522843250242 * \frac{K_{u}}{T_{u}},$ gain on the derivative element is substantially equal to $0.108 * K_{u} * T_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}} * e^{\frac{- 1.4}{K_{u}} + \frac{0.56}{K_{u}^{2}}} - 0.002664037864 {(e^{T_{u}})}^{\log ({1.634220701}^{\log K_{u}})},$ and the reference signal is weighted by a weight substantially equal to $0.25 * e^{\frac{0.56}{K_{u}} + \frac{- 0.12}{K_{u}^{2}}} + \frac{K_{u}}{e^{K_{u}}}$ before the plant output is subtracted therefrom and prior to being fed into the gain element associated with the proportional element.

9. A proportional, integrative, and derivative (PID) controller comprising a proportional element, an integrative element, and a derivative element coupled together and responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, the control signal being substantially the same to that produced by the controller having a proportional element, an integrative element, and a derivative element coupled together and responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, wherein the proportional element has a gain element with a gain being substantially equal to $0.72 * K_{u} * ⅇ$
- -1.6Ku+1.2Ku2-.001234000198⁢
  
  *Tu-6.117274273⁢
  
  *10-6where K_uis the ultimate gain of the plant and T_uis the ultimate period of the plant, and further wherein gain on the integrative element is substantially equal to $\frac{0.72 * K_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}}}{0.59 * T_{u} * e^{\frac{- 1.3}{K_{u}} + \frac{0.38}{K_{u}^{2}}}} - .06852522843250242 * \frac{K_{u}}{T_{u}},$ gain on the derivative element is substantially equal to $0.108 * K_{u} * T_{u} * e^{\frac{- 1.6}{K_{u}} + \frac{1.2}{K_{u}^{2}}} * e^{\frac{- 1.4}{K_{u}} + \frac{0.56}{K_{u}^{2}}} - 0.002664037864 {(e^{T_{u}})}^{\log ({1.634220701}^{\log K_{u}})},$ and the reference signal is weighted by a weight substantially equal to $0.25 * e^{\frac{0.56}{K_{u}} + \frac{- 0.12}{K_{u}^{2}}} + \frac{K_{u}}{e^{K_{u}}}$ before the plant output is subtracted therefrom and prior to being fed into the gain element associated with the proportional element.

10. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller comprises at least one lag element responsive to the A-H output.
- View Dependent Claims (11, 12, 13, 14, 15, 16)
- - 11. The control apparatus defined in claim 10 wherein the subsequent controller comprises at least one input and is operable to generate a control variable that is used as feedback to the input, such that the subsequent controller has internal feedback.
  - 12. The control apparatus defined in claim 10 wherein the subsequent controller comprises at least one input and has an internal feedback loop that provides a signal to the at least one input, the signal being other than an output of the subsequent controller.
  - 13. The control apparatus defined in claim 10 wherein the subsequent controller comprises a subtractor to generate a difference between the A-H output and an output of the subsequent controller.
  - 14. The control apparatus defined in claim 10 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} A (3 + 3 L^{2} E_{13} E_{14} E_{16} E_{17} + L^{3} E_{11} E_{13} E_{14} E_{16} E_{17} + 3 L (1 + \\ E_{16} E_{17})) + (2 + 2 L^{2} E_{13} E_{14} E_{16} E_{17} + L (- 1 + 2 E_{16}) E_{17}) R \end{matrix}}{\begin{matrix} 4 + L^{3} (1 + E_{11}) E_{13} E_{14} E_{16} E_{17} + L^{2} (1 + E_{14} + \\ 2 E_{13} E_{14}) E_{16} E_{17} + 2 L (2 + E_{16} E_{17}) \end{matrix}}$ where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
      
      $L = \frac{1}{1 + T_{r} * s}$ and where;
      
      E₁₁=10^e^{log|log|log(eKu*L)/L||}
      E₁₃=10^e^log|log|K^u^*L||
      E₁₄=e^log|K^u^/L|
      E₁₆=10^e^log|log|K^u^*L||
      and
      E₁₇=e^log(K^u⁾
  - 15. The control apparatus defined in claim 10 wherein the subsequent controller has a transfer function defined as follows:
    - $U = - \frac{- 3 R + A (- 15 - 2 / (1 + E_{25} * s) - 2 / (1 + E_{28} * s))}{\begin{matrix} 16 + \frac{1}{1 + E_{21} * s} - \frac{1}{1 + E_{22} * s} + \frac{1}{1 + E_{23} * s} + \frac{1}{1 + E_{24} * s} + \\ \frac{1}{(1 + E_{25} * s) * (1 + E_{26} * s)} + \frac{1}{(1 + E_{27} * s) * (1 + E_{28} * s)} \end{matrix}}$ where U is an output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
      
      E₂₃=log|2T_r+K_u^log|T^r^+K^u^L^||
      E₂₄=log |T_r+K_u^L|
      E₂₅=|log|T_r+log(T_r+1.2784)||
      E₂₆=^log|T^r^+(T^r⁺⁽x)^log|log|K^u^{^L||})^L|
      where
      x=T_r+K_u^log|log|T^r^+K^u^L^||
      E₂₇=log|T_r+(T_r+K_u^L)^L|
      and
      E₂₈=log|2T_r+K_u^L|
  - 16. The control apparatus defined in claim 10 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} R (1 + 3 E_{34} (1 + E_{32} * s) (1 + E_{33} * s)) + \\ A (10 + E_{34} (3 + E_{31} + 2 E_{32} * s + E_{31} E_{32} * s) (1 + E_{33} * s)) \end{matrix}}{11 + E_{34} (2 + 3 E_{32} * s) (1 + E_{33} * s)}$ where U is the output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
      
      E₃₂=NLM(log|L|−
      
      (abs(L)^L)²T_u³(T_u+1)T_re^L−
      
      2T_ue^L)
      E₃₃=NLM(log|L|−
      
      2T_ue^L(2K_u(log|K_ue^L|−
      
      log|L|)T_u+K_ue^L))
      and
      E₃₄=|log|T_r+1||and where $\begin{matrix} NLM (x) = \begin{matrix} 10^{0} & if x < - 100 or x > 100 \end{matrix} \\ \begin{matrix} 10^{- \frac{100}{19} - \frac{1}{19} x} & if - 100 \leq x < - 5 \end{matrix} \\ \begin{matrix} 10^{\frac{100}{19} - \frac{1}{19} x} & if 5 < x \leq 100 \end{matrix} \\ \begin{matrix} 10^{x} & if - 5 \leq x \leq 5 \end{matrix} \end{matrix}$

17. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller comprises at least one lead element responsive to the A-H output.
- View Dependent Claims (18, 19, 20, 21, 22, 23)
- - 18. The control apparatus defined in claim 17 wherein the subsequent controller comprises at least one input and is operable to generate a control variable that is used as feedback to the input, such that the subsequent controller has internal feedback.
  - 19. The control apparatus defined in claim 17 wherein the subsequent controller comprises at least one input and has an internal feedback loop that provides a signal to the at least one input, the signal being other than an output of the subsequent controller.
  - 20. The control apparatus defined in claim 17 wherein the subsequent controller comprises a subtractor to generate a difference between the A-H output and an output of the subsequent controller.
  - 21. The control apparatus defined in claim 17 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} A (3 + 3 L^{2} E_{13} E_{14} E_{16} E_{17} + L^{3} E_{11} E_{13} E_{14} E_{16} E_{17} + 3 L (1 + E_{16} E_{17})) + \\ (2 + 2 L^{2} E_{13} E_{14} E_{16} E_{17} + L (- 1 + 2 E_{16}) E_{17}) R \end{matrix}}{\begin{matrix} 4 + L^{3} (1 + E_{11}) E_{13} E_{14} E_{16} E_{17} + \\ L^{2} (1 + E_{14} + 2 E_{13} E_{14}) E_{16} E_{17} + 2 L (2 + E_{16} E_{17}) \end{matrix}}$ where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
      
      $L = \frac{1}{1 + T_{r} * s}$ and where;
      
      E₁₁=10^e^{log|log|log(eKu*L)/L||}
      E₁₃=10^e^log|log|K^u^*L||
      E₁₄=e^log|K^u^/L|
      E₁₆=10^e^log|log|K^u^*L||
      and
      E₁₇=e^log(K^u⁾
  - 22. The control apparatus defined in claim 17 wherein the subsequent controller has a transfer function defined as follows:
    - $U = - \frac{- 3 R + A (- 15 - 2 / (1 + E_{25} * s) - 2 / (1 + E_{28} * s))}{\begin{matrix} 16 + \frac{1}{1 + E_{21} * s} - \frac{1}{1 + E_{22} * s} + \frac{1}{1 + E_{23} * s} + \\ \frac{1}{1 + E_{24} * s} + \frac{1}{(1 + E_{25} * s) (1 + E_{26} * s)} + \\ \frac{1}{(1 + E_{27} * s) (1 + E_{28} * s)} \end{matrix}}$ where U is an output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
      
      E₂₃=log|2T_r+K_u^log|T^r^+K^u^L^||
      E₂₄=log|T_r+K_u^L|
      E₂₅=|log|T_r+log(T_r+1.2784)||
      E₂₆=^log|T^r^+(T^r^+(x)^log|log|K^u^{^L||})^L|
      where
      x=T_r+K_u^log|log|T^r^+K^u^L^||
      E₂₇=log|T_r+(T_r+K_u^L)^L|
      and
      E₂₈=log |2T_r+K_u^L|
  - 23. The control apparatus defined in claim 17 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} R (1 + 3 E_{34} (1 + E_{32} * s)) (1 + E_{33} * s)) + A (10 + E_{34} (3 + E_{31} + \\ 2 E_{32} * s + E_{31} E_{32} * s) (1 + E_{33} * s)) \end{matrix}}{11 + E_{34} (2 + 3 E_{32} * s) (1 + E_{33} * s)}$ where U is the output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
      
      E₃₂=NLM(log|L|−
      
      (abs(L)^L)²T_u³(T_u+1)T_re^L−
      
      2T_ue^L)
      E₃₃=NLM(log|L|−
      
      2T_ue^L(2K_u(log|K_ue^L|−
      
      log|L|)T_u+K_ue^L))
      and
      E₃₄=|log|T_r+1||and where $\begin{matrix} NLM (x) = 10^{0} if x < - 100 or x > 100 \\ 10^{- \frac{100}{19} - \frac{1}{19} x} if - 100 \leq x < - 5 \\ 10^{\frac{100}{19} - \frac{1}{19} x} if 5 < x \leq 100 \\ 10^{x} if - 5 \leq x \leq 5. \end{matrix}$

24. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller comprises at least one input and is operable to generate a control variable that is used as feedback to the input, such that the subsequent controller has internal feedback.
- View Dependent Claims (25, 26, 27, 28, 29)
- - 25. The control apparatus defined in claim 24 wherein the subsequent controller comprises at least one input and is operable to generate a control variable that is used as feedback to the input, such that the subsequent controller has internal feedback.
  - 26. The control apparatus defined in claim 24 wherein the subsequent controller comprises at least one input and has an internal feedback loop that provides a signal to the at least one input, the signal being other than an output of the subsequent controller.
  - 27. The control apparatus defined in claim 24 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} A (3 + 3 L^{2} E_{13} E_{14} E_{16} E_{17} + L^{3} E_{11} E_{13} E_{14} E_{16} E_{17} + \\ \begin{matrix} 3 L (1 + E_{16} E_{17})) + (2 + 2 L^{2} E_{13} E_{14} E_{16} E_{17} + \\ L (- 1 + 2 E_{16}) E_{17}) R \end{matrix} \end{matrix}}{\begin{matrix} 4 + L^{3} (1 + E_{11}) E_{13} E_{14} E_{16} E_{17} + L^{2} (1 + E_{14} + \\ 2 E_{13} E_{14}) E_{16} E_{17} + 2 L (2 + E_{16} E_{17}) \end{matrix}}$ where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
      
      $L = \frac{1}{1 + T_{r} * s}$ and where;
      
      E₁₁=10^e^{log|log|log(eKu*L)/L||}
      E₁₃=10^e^log|log|K^u^*L||
      E₁₄=e^log|K^u^/L|
      E₁₆=10^e^log|log|K^u^*L||
      and
      E₁₇=e^log(K^u⁾
  - 28. The control apparatus defined in claim 24 wherein the subsequent controller has a transfer function defined as follows:
    - $U = - \frac{- 3 R + A (- 15 - 2 / (1 + E_{25} * s) - 2 / (1 + E_{28} * s))}{\begin{matrix} 16 + \frac{1}{1 + E_{21} * s} - \frac{1}{1 + E_{22} * s} + \frac{1}{1 + E_{23} * s} + \\ \frac{1}{1 + E_{24} * s} + \frac{1}{(1 + E_{25} * s) (1 + E_{26} * s)} + \\ \frac{1}{(1 + E_{27} * s) (1 + E_{28} * s)} \end{matrix}}$ where U is an output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
      
      E₂₃=log|2T_r+K_u^log|T^r^+K^u^L||
      E₂₄=log|T_r+K_u^L|
      E₂₅=|log|T_r+log(T_r+1.2784)||
      E₂₆=log|T_r+(T_r+(x)^log|log|K^u^*L||)^L|
      where
      x=T_r+K_u^log|log|T^r^+K^u^||
      E₂₇=log|T_r+(T_r+K_u^L)^L|
      and
      E₂₈=log|2T_r+K_u^L|
  - 29. The control apparatus defined in claim 24 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} R (1 + 3 E_{34} (1 + E_{32} * s) (1 + E_{33} * s)) + \\ A (10 + E_{34} (3 + E_{31} + 2 E_{32} * s + E_{31} E_{32} * s) (1 + E_{33} * s)) \end{matrix}}{11 + E_{34} (2 + 3 E_{32} * s) (1 + E_{33} * s)}$ where U is the output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
      
      E₃₂=NLM(log|L|−
      
      (abs(L)^L)²T_u³(T_u+1)T_re^L−
      
      2T_ue^L)
      E₃₃=NLM(log|L|−
      
      2T_ue^L(2K_u(log|K_ue^L|−
      
      log|L|)T_u+K_ue^L))
      and
      E₃₄=|log|T_r+1||and where $\begin{matrix} NLM (x) = \begin{matrix} 10^{0} & if x < - 100 or x > 100 \end{matrix} \\ \begin{matrix} 10^{- \frac{100}{19} - \frac{1}{19} x} & if - 100 \leq x < - 5 \end{matrix} \\ \begin{matrix} 10^{\frac{100}{19} - \frac{1}{19} x} & if 5 < x \leq 100 \end{matrix} \\ \begin{matrix} 10^{x} & if - 5 \leq x \leq 5 \end{matrix} \end{matrix}$

30. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller comprises at least one input and has an internal feedback loop that provides a signal to the at least one input, the signal being other than an output of the subsequent controller.
- View Dependent Claims (31, 32, 33, 34, 35)
- - 31. The control apparatus defined in claim 30 wherein the subsequent controller comprises at least one input and is operable to generate a control variable that is used as feedback to the input, such that the subsequent controller has internal feedback.
  - 32. The control apparatus defined in claim 30 wherein the subsequent controller comprises at least one input and has an internal feedback loop that provides a signal to the at least one input, the signal being other than an output of the subsequent controller.
  - 33. The control apparatus defined in claim 30 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} A (3 + 3 L^{2} E_{13} E_{14} E_{16} E_{17} + L^{3} E_{11} E_{13} E_{14} E_{16} E_{17} + 3 L (1 + \\ E_{16} E_{17})) + (2 + 2 L^{2} E_{13} E_{14} E_{16} E_{17} + L (- 1 + 2 E_{16}) E_{17}) R \end{matrix}}{\begin{matrix} 4 + L^{3} (1 + E_{11}) E_{13} E_{14} E_{16} E_{17} + L^{2} (1 + E_{14} + \\ 2 E_{13} E_{14}) E_{16} E_{17} + 2 L (2 + E_{16} E_{17}) \end{matrix}}$ where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
      
      $L = \frac{1}{1 + T_{r} * s}$ and where;
      
      E₁₁=10^e^{log|log|log(eKu*L)/L||}
      E₁₃=10^e^log|log|K^u^*L||
      
      E₁₄=e^log|K^u^/L|
      E₁₆=10^e^log|log|K^u^*L||
      and
      E₁₇=e^log(K^u⁾
  - 34. The control apparatus defined in claim 30 wherein the subsequent controller has a transfer function defined as follows:
    - $U = - \frac{- 3 R + A (- 15 - 2 / (1 + E_{25} * s) - 2 / (1 + E_{28} * s))}{\begin{matrix} 16 + \frac{1}{1 + E_{21} * s} - \frac{1}{1 + E_{22} * s} + \frac{1}{1 + E_{23} * s} + \frac{1}{1 + E_{24} * s} + \\ \frac{1}{(1 + E_{25} * s) (1 + E_{26} * s)} + \frac{1}{(1 + E_{27} * s) (1 + E_{28} * s)} \end{matrix}}$ where U is an output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} - T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
      
      E₂₃=log|2T_r+K_u^log|T^r^+K^u^L^||
      E₂₄=log|T_r+K_u^L|
      E₂₅=|log|T_r+log(T_r+1.2784)||
      
      E₂₆=^log|T^r^+(T^r^+(x)^log|log|K^u^{^L||}⁾^L|
      where
      x=T_r+K_u^log|log|T^r^+K^u^L||
      E₂₇=log|T_r+(T_r+K_u^L)^L|
      and
      E₂₈=log|2T_r+K_u^L|
  - 35. The control apparatus defined in claim 30 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} R (1 + 3 E_{34} (1 + E_{32} * s) (1 + E_{33} * s)) + \\ A (10 + E_{34} (3 + E_{31} + 2 E_{32} * s + E_{31} E_{32} * s) (1 + E_{33} * s)) \end{matrix}}{11 + E_{34} (2 + 3 E_{32} * s) (1 + E_{33} * s)}$ where U is the output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
      
      E₃₂=NLM(log|L|−
      
      (abs(L)^L)²T_u³(T_u+1)T_re^L−
      
      2T_ue^L)
      E₃₃=NLM(log|L|−
      
      2T_ue^L(2K_u(log|K_ue^L|−
      
      log|L|)T_u+K_ue^L))
      and
      
      E₃₄=|log|T_r+1||and where $\begin{matrix} NLM (x) = \begin{matrix} 10^{0} & if x < - 100 or x > 100 \end{matrix} \\ \begin{matrix} 10^{- \frac{100}{19} - \frac{1}{19} x} & if - 100 \leq x < - 5 \end{matrix} \\ \begin{matrix} 10^{\frac{100}{19} - \frac{1}{19} x} & if 5 < x \leq 100 \end{matrix} \\ \begin{matrix} 10^{x} & if - 5 \leq x \leq 5 \end{matrix} \end{matrix}$

36. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller comprises a subtractor to generate a difference between the A-H output and an output of the subsequent controller.
- View Dependent Claims (37, 38, 39, 40, 41)
- - 37. The control apparatus defined in claim 36 wherein the subsequent controller comprises at least one input and is operable to generate a control variable that is used as feedback to the input, such that the subsequent controller has internal feedback.
  - 38. The control apparatus defined in claim 36 wherein the subsequent controller comprises at least one input and has an internal feedback loop that provides a signal to the at least one input, the signal being other than an output of the subsequent controller.
  - 39. The control apparatus defined in claim 36 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} A (3 + 3 L^{2} E_{13} E_{14} E_{16} E_{17} + L^{3} E_{11} E_{13} E_{14} E_{16} E_{17} + \\ \begin{matrix} 3 L (1 + E_{16} E_{17})) + (2 + 2 L^{2} E_{13} E_{14} E_{16} E_{17} + \\ L (- 1 + 2 E_{16}) E_{17}) R \end{matrix} \end{matrix}}{\begin{matrix} 4 + L^{3} (1 + E_{11}) E_{13} E_{14} E_{16} E_{17} + L^{2} (1 + E_{14} + \\ 2 E_{13} E_{14}) E_{16} E_{17} + 2 L (2 + E_{16} E_{17}) \end{matrix}}$ where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
      
      $\begin{matrix} L = \frac{1}{1 + T_{r} * s} \end{matrix}$ and where;
      
      E₁₁=10^e^{log|log|log(eKu*L)/L||}
      E₁₃=10^e^log|log|K^u^*L||
      E₁₄=e^log|K^u^/L|
      E₁₆=10^e^log|log|K^u^*L||
      and
      E₁₇=e^log(K^u⁾.
  - 40. The control apparatus defined in claim 36 wherein the subsequent controller has a transfer function defined as follows:
    - $U = - \frac{- 3 R + A (- 15 - 2 / (1 + E_{25} * s) - 2 / (1 + E_{28} * s))}{\begin{matrix} 16 + \frac{1}{1 + E_{21} * s} - \frac{1}{1 + E_{22} * s} + \frac{1}{1 + E_{23} * s} + \\ \frac{1}{1 + E_{24} * s} + \frac{1}{(1 + E_{25} * s) (1 + E_{26} * s)} + \\ \frac{1}{(1 + E_{27} * s) (1 + E_{28} * s)} \end{matrix}}$ where U is an output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
      
      E₂₃=log|2T_r+K_u^log|T^r^+K^u^L^L||
      E₂₄=log|T_r+K_u^L|
      E₂₅=|log|T_r+log(T_r+1.2784)||
      E₂₆=^log|T^r+(T^r^+(x)^log|log|K^u^{^L||}⁾^L|where
      x=T_r+K_u^log|log|T^r^+K^u^L||
      E₂₇=log|T_r+(T_r+K_u^L)^L|
      and
      E₂₈=log|2T_r+K_u^L|
  - 41. The control apparatus defined in claim 36 wherein the subsequent controller has a transfer function defined as follows:
    - $U = \frac{\begin{matrix} R (1 + 3 E_{34} (1 + E_{32} * s) (1 + E_{33} * s)) + A (10 + E_{34} (3 + E_{31} + \\ 2 E_{32} * s + E_{31} E_{32} * s) (1 + E_{33} * s)) \end{matrix}}{11 + E_{34} (2 + 3 E_{32} * s) (1 + E_{33} * s)}$ where U is the output of the subsequent controller, A is the output of the Astrom-Hagglund controller, R is a reference signal, and where;
      
      $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
      
      E₃₂=NLM(log|L|−
      
      (abs(L)^L)²T_u³(T_u+1)T_re^L−
      
      2T_ue^L)
      E₃₃=NLM(log|L|−
      
      2T_ue^L(2K_u(log|K_ue^L|−
      
      log|L|)T_u+K_ue^L))
      and
      E₃₄=|log|T_r+1||and where $\begin{matrix} NLM (x) = 10^{0} if x < - 100 or x > 100 \\ 10^{\cdot \frac{100}{19} \cdot \frac{1}{19} x} if - 100 \leq x < - 5 \\ 10^{\cdot \frac{100}{19} \cdot \frac{1}{19} x} if 5 < x \leq 100 \\ 10^{x} if - 5 \leq x \leq 5 \end{matrix}$

42. A controller comprising lag elements and proportional, derivative, integrative, adder, and subtractor elements whose output is the sum of two signals, the elements coupled together to generate a controller output in response to a reference signalthe first of these two signals being a result of subtracting a result of applying a gain of 3 to the controller output from a result of applying a gain of 3 to the output of a Astrom-Hagglund controller, the second of these two signals being a sum of a result of applying a gain of 2 to a reference signal and the result of applying a lag parameterized by
T_rto a incoming signal, where T_ris the time constant of the plant,said incoming signal being a result of subtracting the second of the two signals from a result of applying a gain of
e^log(K^u⁾, where K_uis the ultimate gain of the plant to the result of subtracting the reference signal from a result of applying gain of
10^e^log|log|K^u^*L||to the sum of two earlier signals,the first of the two earlier signals being the result of a three-way subtraction in which the result of applying a gain of 2 to the controller output is subtracted from a sum of a result of applying gain of 3 to an output of the Astrom-Hagglund controller and a result of applying gain of 2 to the reference signal, the second of the two earlier signals being the result of applying a lag parameterized by
- T_rto a result of subtracting the controller output from a result of applying a gain of
  e^log|K^/L|, where L is the dead time of the plant, to the result of subtracting the controller output from the result of applying gain of
  10^e^log|log|K^u^*L||to a result of three-way subtraction in which a result of applying a gain of 2 to the controller output is subtracted from a sum of the result of applying a gain of 3 to the output of the Astrom-Hagglund controller and a sum of two intermediate signals, the first of the two intermediate signals being the result of applying a gain of 2 to the reference signal, the second of the two intermediate signals being the result of applying a lag parameterized by
  T_rto a result of subtracting the controller output from the result of applying a gain of
  10^e^{log|log|log(eKu*L)/L||}to the result of subtracting the controller output from the output of the Astrom-Hagglund controller.

43. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller has a transfer function given by the equation $U = \frac{\begin{matrix} A (3 + 3 L^{2} E_{13} E_{14} E_{16} E_{17} + L^{3} E_{11} E_{13} E_{14} E_{16} E_{17} + \\ \begin{matrix} 3 L (1 + E_{16} E_{17})) + (2 + 2 L^{2} E_{13} E_{14} E_{16} E_{17} + \\ L (- 1 + 2 E_{16}) E_{17}) R \end{matrix} \end{matrix}}{\begin{matrix} 4 + L^{3} (1 + E_{11}) E_{13} E_{14} E_{16} E_{17} + L^{2} (1 + E_{14} + \\ 2 E_{13} E_{14}) E_{16} E_{17} + 2 L (2 + E_{16} E_{17}) \end{matrix}}$ where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
  
  $\begin{matrix} L = \frac{1}{1 + T_{r} * s} \end{matrix}$ and where;
  
  E₁₁=10^e^{log|log|log(eKu*L)/L||}
  E₁₃=10^e^log|log|K^u^*L||
  E₁₄=e^{log |K}^u^/L|
  E₁₆=10^e^log|log|K^u^*L||
  and
  E₁₇=e^log(K^u^).

44. A control apparatus responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, the control signal being substantially the same to that produced by the controller having an Astrom-Hagglund (A-H) controller, operable to generate an A-H output, and a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller has a transfer function given by the following equation $U = \begin{matrix} A (3 + 3 \end{matrix}$
- L2⁢
  
  E13⁢
  
  E14⁢
  
  E16⁢
  
  E17+L3⁢
  
  E11⁢
  
  E13⁢
  
  E14⁢
  
  E16⁢
  
  E17+3⁢
  
  L⁢
  
  (1+E16⁢
  
  E17))+(2+2⁢
  
  L2⁢
  
  E13⁢
  
  E14⁢
  
  E16⁢
  
  E17+L⁢
  
  (-1+2⁢
  
  E16)⁢
  
  E17)⁢
  
  R4+L3⁡
  
  (1+E11)⁢
  
  E13⁢
  
  E14⁢
  
  E16⁢
  
  E17+L2(1+E14+2⁢
  
  E13⁢
  
  E14)⁢
  
  E16⁢
  
  E17+2⁢
  
  L⁡
  
  (2+E16⁢
  
  E17)where U is a controller output, A is a Astrom-Hagglund controller output, R is a reference signal, and L is a transfer function for the lag blocks used in the controller;
  
  $L = \frac{1}{1 + T_{r} * s}$ and where;
  
  E₁₁=10^e^{log|log|log(eKu*L)/L||}
  E₁₃=10^e^log|log|K^u^*L||
  E₁₄=e^log|K^u^/L|
  E₁₆=10^e^log|log|K^u^*L||
  and
  
  E₁₇=e^log(K^u^).

45. A control apparatus to generate a control variable output, the control apparatus comprising:
- an Astrom-Hagglund (A-H) controller having an A-H output responsive to a reference signal and a plant output;
  
  a first gain block to apply a gain of three to the reference signal to generate a first gain output;
  
  a first subtractor to subtract the control variable output from the A-H output to generate a first subtractor output;
  
  a second gain block to apply a gain of fifteen to the first subtractor output to generate a second gain output;
  
  a first adder to add the first and second gain outputs to generate a first adder output;
  
  a first lag block to apply a lag parameterized by $abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$ to the control variable output to generate a first lag output, where K_uis the ultimate gain of the plant, where T_uis the ultimate period of the plant, where L is the dead time of the plant, and where T_ris the time constant of the plant;
  
  a second lag block to apply a lag parameterized by
  log|2T_r+K_u^L|to the control variable output to generate a second lag output;
  
  a second subtractor to subtract the second lag output from the first lag output to generate a second subtractor output;
  
  a third lag block to apply a lag parameterized by
  log|2T_r+K_u^log|T^r^+K^L||to the control variable output to generate a third lag output;
  
  a third subtractor to subtract the third lag output from the second subtractor output to generate a third subtractor output;
  
  a fourth lag block to apply a lag parameterized by
  log|T_r+K_u^L|to the control variable output to generate a fourth lag output;
  
  a fourth subtractor to subtract the fourth lag output from the third subtractor output to generate a fourth subtractor output;
  
  a third gain block to apply a gain of two to the A-H output to generate a third gain output;
  
  a fifth lag block to apply a lag parameterized by
  log|T_r+(T_r+(x)^log|log|K^u^{^L||})^L|
  where
  x=T_r+K_u^log|log|T^r^+K^u^L||to the control variable output to generate a fifth lag output;
  
  a fifth subtractor to subtract the fifth lag output from the third gain output to generate a fifth subtractor output;
  
  a sixth lag block to apply a lag parameterized by
  |log|T_r+log(T_r+1.2784)||to the fifth subtractor output to generate a sixth lag output;
  
  a seventh lag block to apply a lag parameterized by
  log|T_r+(T_r+K_u^L)^L|to the control variable output to generate a seventh lag output;
  
  a sixth subtractor to subtract the seventh lag output from the third gain output to generate a sixth subtractor output;
  
  a eighth lag block to apply a lag pararneterized by
  log|2T_r+K_u^L|to the sixth subtractor output to generate a eighth lag output;
  
  a second adder to add the sixth lag output and the eighth lag output to generate a second adder output; and
  
  a third adder to add the first adder output, the fourth subtractor output and the eighth lag output to generate the control variable output.

46. A control apparatus comprising:
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller has a transfer function given by the following equation $U = - \frac{- 3 R + A (- 15 - 2 / (1 + E_{25} * s) - 2 / (1 + E_{28} * s))}{\begin{matrix} 16 + \frac{1}{1 + E_{21} * s} - \frac{1}{1 + E_{22} * s} + \frac{1}{1 + E_{23} * s} + \\ \frac{1}{1 + E_{24} * s} + \frac{1}{(1 + E_{25} * s) (1 + E_{26} * s)} + \\ \frac{1}{(1 + E_{27} * s) (1 + E_{28} * s)} \end{matrix}}$ where U is the output of the controller, A is the output of the Astrom-Hagglund controller, R is the reference signal, and where;
  
  E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
  
  E₂₃=log|2T_r+K_u^log|T^+K^u^L||
  E₂₄=log|T_r+K_u^L|
  E₂₅=|log|T_r+log(T_r+1.2784)||
  E₂₆=^log|T^r^+(T^r^+(x)^log|log|K^u^{^L||}⁾^L|
  where
  x=T_r+K_u^log|log|T^r^+K^u^L||
  E₂₇=log|T_r+(T_r+K_u^L)^L|
  and
  E₂₈=log|2T_r+K_u^L|

47. A control apparatus responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, the control signal being substantially the same to that produced by the controller having an Astrom-Hagglund (A-H) controller, operable to generate an A-H output, and a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller has a transfer function given by the following equation $U = - - 3$
- R+A⁡
  
  (-15-2/(1+E25*s)-2/(1+E28*s))16+11+E21*s-11+E22*s+11+E23*s+11+E24*s+1(1+E25*s)⁢
  
  (1+E26*s)+1(1+E27*s)⁢
  
  (1+E28*s)where U is the output of the controller, A is the output of the Astrom-Hagglund controller, R is the reference signal, and where;
  
  E₂₁=log|2T_r+K_u^L| $E_{22} = abs (\log (abs (T_{r} + K_{u}^^{K_{u}^^{(\frac{0.68631}{\frac{1}{\frac{K_{u}^^{\frac{0.13031}{\frac{T_{u}}{\log \langle \log \langle T_{r} + {K_{u}}^{(K_{u}^{L})} \rangle \rangle} \cdot {(K_{u}^{L})}^{L} \cdot T_{u}}}}{\frac{1}{T_{r} + K_{u}^^{\frac{0.69897}{T_{r} + K_{u}^{L}}}} \cdot T_{r}^{L} - T_{u}}} \cdot T_{r}^{L} \cdot T_{u}})}})))$
  
  E₂₃=log|2T_r+K_u^log|T^r^+K^u^L||
  E₂₄=log|T_r+K_u^L|
  E₂₅=log|T_r+log(T_r+1.2784)||
  
  E₂₆=^log|T^+(T^r^+(x)^log|log|k^u^{^L||}⁾^L|
  where
  x=T_r+K_u^log|log|T^r^+K^u^L^||
  and
  E₂₈=log |2T_r+K_u^L|

48. A control apparatus to generate a control variable output, the control apparatus comprising:
- an Astrom-Hagglund (A-H) controller having an A-H output responsive to a reference signal and a plant output;
  
  a first gain block to apply a gain of 3 to the reference signal to generate a first gain output;
  
  a second gain block to apply a gain of $\langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$ to the A-H output to generate a second gain output, where T_uis the ultimate period of the plant, where L is the dead time of the plant, and where T_ris the time constant of the plant;
  
  a first subtractor to subtract the control variable output from the second gain output to generate a first subtractor output;
  
  a third gain block to apply a gain of 2 to the A-H output to generate a third gain output;
  
  a fourth gain block to apply a gain of 2 to the control variable output to generate a fourth gain output;
  
  a second subtractor to subtract the fourth gain output from the third gain output to generate a second subtractor output;
  
  a first adder to add the first gain output, the first subtractor output and the second subtractor output to generate a first adder output;
  
  a first lead block to apply a lead parameterized by
  NLM(log|L|−
  
  (abs(L)^L)²T_u³(T_u+1)T_e^L−
  
  2T_ue^L) to the first adder output to generate a first lead output;
  
  a second adder to add the first lead output, the A-H output and the control variable output to generate a second adder output;
  
  a second lead block to apply a lead parameterized by
  NLM(log|L|−
  
  2T_ue^L(2K_u(log|K_ue^L|−
  
  log|L|)T_u+K_ue^L)), to the second adder output to generate a second lead output, where K_uis the ultimate gain of the plant;
  
  a fifth gain block to apply a gain of
  |log|T^r+1||, to the second lead output to generate a fifth gain output;
  
  a third subtractor to subtract the control variable output from the A-H output to generate a third subtractor output;
  
  a sixth gain block to apply a gain of ten to the third subtractor output to generate a sixth gain output; and
  
  a third adder to add the reference signal, the fifth gain output, and the sixth gain output to generate the control variable output.

49. A control apparatus comprising;
- an Astrom-Hagglund (A-H) controller operable to generate an A-H output;
  
  a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller has a transfer function given by the following equation $U = \frac{\begin{matrix} R (1 + 3 E_{34} (1 + E_{32} * s) (1 + E_{33} * s)) + A (10 + E_{34} (3 + E_{31} + \\ 2 E_{32} * s + E_{31} E_{32} * s) (1 + E_{33} * s)) \end{matrix}}{11 + E_{34} (2 + 3 E_{32} * s) (1 + E_{33} * s)}$ where U is an output of the subsequent controller, A is an output of the Astrom-Hagglund controller, R is a reference signal, and where $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
  
  E₃₂=NLM(log|L|−
  
  (abs(L)^L)²T_u³(T_u+1)T_re^L−
  
  2T_ue^L)
  E₃₃=NLM(log|L|−
  
  2T_ue^L(2K_u(log|K_ue^L|−
  
  log|L|)T_u+K_ue^L))
  and
  E₃₄|log|T_r+1||and where $\begin{matrix} NLM (x) = 10^{0} if x < - 100 or x > 100 \\ 10^{\frac{100}{19} - \frac{1}{19} x} if - 100 \leq x < - 5 \\ 10^{- \frac{100}{19} - \frac{1}{19} x} if 5 < x \leq 100 \\ 10^{x} if - 5 \leq x \leq 5. \end{matrix}$

50. A control apparatus responsive to a reference signal to generate a control signal in response thereto to cause a plant to generate a plant output, the control signal being substantially the same to that produced by the controller having an Astrom-Hagglund (A-H) controller, operable to generate an A-H output, and a subsequent controller operable in response to the A-H output to generate a control variable to control operation of a plant, wherein the subsequent controller has a transfer function given by the following equation $U = \begin{matrix} R \end{matrix}$
- (1+3⁢
  
  E34⁡
  
  (1+E32*s)⁢
  
  (1+E33*s))+A(10+E34(3+E31+2⁢
  
  E32*s+E31⁢
  
  E32*s)⁢
  
  (1+E33*s))11+E34⁡
  
  (2+3⁢
  
  E32*s)⁢
  
  (1+E33*s)where U is an output of the subsequent controller, A is an output of the Astrom-Hagglund controller, R is a reference signal, and where $E_{31} = \langle \log \langle T_{r} - T_{u} + \log \langle \frac{\log ({\langle L \rangle}^{L})}{T_{u} + 1} \rangle \rangle \rangle$
  
  E₃₂=NLM(log|L|−
  
  (abs(L)^L)²T_u³(T_u+1)T_re^L−
  
  2T_ue^L)
  E₃₃=NLM(log|L|−
  
  2T_ue^L(2K_u(log|K_ue^L|−
  
  log|L|)T_u+K_ue^L))
  and
  E₃₄=|log|T_r+1||and where $\begin{matrix} NLM (x) = 10^{0} if x < - 100 or x > 100 \\ 10^{- \frac{100}{19} - \frac{1}{19} x} if - 100 \leq x < - 5 \\ 10^{\frac{100}{19} - \frac{1}{19} x} if 5 < x \leq 100 \\ 10^{x} if - 5 \leq x \leq 5 \end{matrix}$

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Current Assignee
John R. Koza
Original Assignee
John R. Koza
Inventors
Streeter, Matthew J., Keane, Martin A., Koza, John R.
Primary Examiner(s)
PATEL, RAMESH B

Application Number

US10/194,190
Time in Patent Office

928 Days
Field of Search

700/28, 700/32, 700/37, 700/39, 700 40- 41, 700/42, 318/609, 318/610
US Class Current

700/42
CPC Class Codes

G05B 11/42 for obtaining a characteris...

Apparatus for improved general-purpose PID and non-PID controllers

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Apparatus for improved general-purpose PID and non-PID controllers

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