**5. Controllers for first order system**

As a first order system is considered, this leads to the one block controller architecture only.

#### **5.1 P controller**

Here we have

(16)

2 *x*

*u s =C s x s x s* () () () () 1 1 *<sup>r</sup>* (17)

1 *x*

36 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

This is the cascade controller architecture. For 2nd order system the cascade controller

2 22 11 22 11

*u*

( ) ( ) {( ) ( )} *<sup>r</sup> <sup>r</sup> v r pr <sup>C</sup> us C x x x x C x x C x x*

Figure 2 presents the block diagram of the cascade controller architecture for a 2nd order system. The rationale for the notation of *Cp* (position) and *Cv* (velocity) will be presented in

By elementary operation on (13), and exploiting the relations between the state space

*x*1*<sup>r</sup> x*<sup>1</sup> *u*

Although from input-output transfer function point-of-view, there is no formal difference between the different architectures, there is difference with respect to the response to initial

conditions, effects of saturation and nonlinearities, robustness, and more.

<sup>1</sup>

Fig. 2. Cascade controller architecture for 2nd order system.

variables, the one block controller architecture can be written as

Figure 3 presents the block diagram of the one block controller architecture.

2

*Cp Cv*

*C* 

architecture takes the form.

*r* 2 *x*

*r*1 *x*

**4.3 One block controller architecture** 

Fig. 3. One block controller architecture.

**4.4 Discussion** 

the sequel.

$$
\mu = k\_1 e\_\times = k\_1 \left(\mathbf{x}\_r - \mathbf{x}\right) \tag{18}
$$

This is the proportional - P controller.

$$\mathbf{C}(\mathbf{s}) = k\_1 \tag{19}$$

#### **5.2 PI controller**

Here we have

$$\begin{aligned} \dot{\eta}\_1 &= e\_\mathbf{x} \\ \mu &= R^{-1} B^T \begin{bmatrix} P\_{11} & P\_{12} \end{bmatrix} \begin{bmatrix} e\_\mathbf{x} \\ \eta \end{bmatrix} = \begin{bmatrix} k\_1 & k\_2 \end{bmatrix} \begin{bmatrix} e\_\mathbf{x} \\ \eta\_1 \end{bmatrix} = k\_1 e\_\mathbf{x} + k\_2 \int e\_\mathbf{x} dt \end{aligned} \tag{20}$$

This is the proportional + Integral - PI controller.

$$\text{CC}(s) = k\_1 + \frac{k\_2}{s} = \frac{k\_1s + k\_2}{s} \tag{21}$$

**5.3 PI<sup>2</sup> controller** 

Here we have

$$\begin{aligned} \dot{\eta}\_1 &= e\_\mathbf{x} \\ \dot{\eta}\_2 &= \eta\_{1'} \text{ or } \ddot{\eta}\_{-2} = e\_\mathbf{x} \end{aligned} \tag{22}$$

$$\begin{aligned} \dot{\eta} &= \mathbf{R}^{-1} \mathbf{B}^T \begin{bmatrix} P\_{11} & P\_{12} \end{bmatrix} \begin{bmatrix} e\_\mathbf{x} \\ \eta \end{bmatrix} = \begin{bmatrix} k\_1 & k\_{21} & k\_{22} \end{bmatrix} \begin{bmatrix} e\_\mathbf{x} \\ \eta\_1 \\ \eta\_2 \end{bmatrix} = k\_1 e\_\mathbf{x} + k\_{21} \int e\_\mathbf{x} dt + k\_{22} \iint e\_\mathbf{x} dt \end{aligned} \tag{23}$$

This is the proportional + double integrator - PI2 controller.

$$C(s) = k\_1 + \frac{k\_{21}}{s} + \frac{k\_{22}}{s^2} = \frac{k\_1s^2 + k\_{21}s + k\_{22}}{s^2} \tag{23}$$

**5.4 PIm controller** 

Here we have

$$\begin{aligned} \dot{\eta}\_1 &= e\_\chi\\ \dot{\eta}\_2 &= \eta\_{1\prime\prime} \text{ or } \ddot{\eta}\_{\;\;2} = e\_\chi\\ \vdots\\ \dot{\eta}\_m &= \eta\_{m-1\,,\;} \text{ or } \eta\_m^{(m)} = e\_\chi \end{aligned} \tag{24}$$

Family of the PID Controllers 39

22 2 2

The reason for selecting the state space representation (27) is that plant without zero, i.e. <sup>1</sup> *b* 0 , is a case that is often met in motion control with electrical and PZT motors (Rusnak,

*x y x y xxx x* 

; *r r r r*

; *r r r r*

(31)

(32)

(33)

(34)

*uk k* 1 r 2 2r 2 y -y x -x (35)

(36)

*D*

(37)

*D*

1 1

1 1 2 2

Feedback without integral action is implemented. The tracking errors are

*x x*

*e*

*x y x y x y x y* 

and one can deal with position feedback, feedback on *y* , and velocity feedback, feedback on *y* . For this reason in this chapter we will call, with slight abuse of nomenclature, the feedback loop on *y* the position loop and the feedback loop on 2 *x y* ,( ), the velocity loop.

1 r 1r 1

y -y x -x

*e e*

 <sup>1</sup> 1 2 1 2 2r 2 2 = x -x *<sup>x</sup> x e u k k ke k e* 

> <sup>1</sup> 2 2r 2 r

*<sup>k</sup> u k*

( ) x -x x -x

To get the one block controller we substitute (30) and get (in Laplace domain)

*k x u s = k e k k e*

1 2

2 x -x y -y

*k*

 2 2 1 2 2r 2 1 1r 1

2 2

( ) ( ) ( ) ( ) <sup>1</sup>

*u s bs ab k s Cs = k k <sup>k</sup> es bs b ab s*

1 2 11

1

*s x*

*P*

2 2r 2

x -x

2000a). For plant without zero 2 *x y* , so that

**6.1 PD controller** 

The controller is

**6.1.1 Parallel controller** 

**6.1.2 Cascade controller** 

**6.1.3 One block controller** 

$$\mu = \mathbf{R}^{-1} \mathbf{B}^{T} \begin{bmatrix} P\_{11} & P\_{12} \end{bmatrix} \begin{bmatrix} e\_{\mathbf{x}} \\ \eta \end{bmatrix} = \begin{bmatrix} k\_{1} & k\_{21} & k\_{22} & \cdots & k\_{2m} \end{bmatrix} \begin{bmatrix} e\_{\mathbf{x}} \\ \eta\_{1} \\ \eta\_{2} \\ \vdots \\ \eta\_{m} \end{bmatrix} \tag{25}$$
 
$$= k\_{1}e\_{\mathbf{x}} + k\_{21} \int e\_{\mathbf{x}} d\tau + \ldots + k\_{2m} \int \cdots \int e\_{\mathbf{x}} d\tau\_{1} \cdots d\tau\_{m}$$

This is the proportional + (m) integrators - PIm controller.

$$\text{C(s)} = k\_1 + \frac{k\_{21}}{s} + \dots + \frac{k\_{2m}}{s^m} = \frac{k\_1 s^m + k\_{21} s^{m-1} + \dots + k\_{2m}}{s^m} \tag{26}$$

Table 1 summarizes the one block generalized PID controller structure for first order system.


Table 1. One block generalized PID controllers for 1st order system.

### **6. Controllers for second order system**

Second order plant and the trajectory generator are assumed and are represented in the companion form

$$A = A\_r = \begin{bmatrix} 0 & 1 \\ -a\_2 & -a\_1 \end{bmatrix}, B = \begin{bmatrix} b\_1 \\ b\_2 \end{bmatrix}, \ C = C\_r = \begin{bmatrix} 1 & 0 \end{bmatrix}, \tag{27}$$

and we have

$$H(\mathbf{s}) = H\_r(\mathbf{s}) = \frac{\mathbf{x}\_1}{\mu} = \frac{\mathbf{y}}{\mu} = \frac{b\_1 \mathbf{s} + (b\_2 + a\_1 b\_1)}{\mathbf{s}^2 + a\_1 \mathbf{s} + a\_2},\tag{28}$$

$$\frac{x\_2}{u} = \frac{b\_2s - a\_2b\_1}{s^2 + a\_1s + a\_2} \tag{29}$$

$$\frac{x\_2}{x\_1} = \frac{b\_2s - a\_2b}{b\_1s + (b\_2 + a\_1b\_1)}\tag{30}$$

The plant's and trajectory's state generator are denoted

21 2 1 21 2

*m m*

1 21 2 1

controller P 1 *k* PI 1 2 *ks k*

PI2 <sup>2</sup>

2 1 0 1

Table 1. One block generalized PID controllers for 1st order system.

*A Ar a a* 

The plant's and trajectory's state generator are denoted

**6. Controllers for second order system** 

companion form

and we have

*ke k ed k ed d*

*e u =R B P P k k k k*

*x x mx m*

*<sup>k</sup> k ks k s k C s = k s s s*

Table 1 summarizes the one block generalized PID controller structure for first order system.

*s* 

Second order plant and the trajectory generator are assumed and are represented in the

, <sup>1</sup> 2 *b*

( ) () () , *<sup>r</sup> x bs b ab <sup>y</sup> Hs H s*

> 2 2 21 2

*x bs ab u s as a* 

2 22 1 1 2 11 ( ) *x bs ab x bs b ab* 

*b* 

1 1 2 11 2

*u u s as a*

1 2

1 2

*B*

PIm <sup>1</sup>

1 21 22 2 *ks k s k s* 

1 21 2 *m m*

*m ks k s k s* 

*m*

, *C C <sup>r</sup>* 1 0 , (27)

*T x*

11 12 1 21 22 2 2

*m m m m*

 

1

 (26)

1

This is the proportional + (m) integrators - PIm controller.

<sup>1</sup> ( )

1

*m*

*m*

*x*

*e*

(25)

(28)

(29)

(30)

$$
\begin{bmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{bmatrix} = \begin{bmatrix} \mathbf{y} \\ \mathbf{x}\_2 \end{bmatrix}; \begin{bmatrix} \mathbf{x}\_{r1} \\ \mathbf{x}\_{r2} \end{bmatrix} = \begin{bmatrix} \mathbf{y}\_r \\ \mathbf{x}\_{r2} \end{bmatrix} \tag{31}
$$

The reason for selecting the state space representation (27) is that plant without zero, i.e. <sup>1</sup> *b* 0 , is a case that is often met in motion control with electrical and PZT motors (Rusnak, 2000a). For plant without zero 2 *x y* , so that

$$
\begin{bmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{bmatrix} = \begin{bmatrix} \mathbf{y} \\ \dot{\mathbf{y}} \end{bmatrix}; \begin{bmatrix} \mathbf{x}\_{r1} \\ \mathbf{x}\_{r2} \end{bmatrix} = \begin{bmatrix} \mathbf{y}\_r \\ \dot{\mathbf{y}}\_r \end{bmatrix} \tag{32}
$$

and one can deal with position feedback, feedback on *y* , and velocity feedback, feedback on *y* . For this reason in this chapter we will call, with slight abuse of nomenclature, the feedback loop on *y* the position loop and the feedback loop on 2 *x y* ,( ), the velocity loop.

#### **6.1 PD controller**

Feedback without integral action is implemented. The tracking errors are

$$\begin{aligned} \mathbf{e}\_{x1} &= \mathbf{y}\_{\text{r}} \text{-} \mathbf{y} &= \mathbf{x}\_{1\text{r}} \text{-} \mathbf{x}\_{1} = \mathbf{e} \\ \mathbf{e}\_{x2} &= \mathbf{x}\_{2\text{r}} \text{-} \mathbf{x}\_{2} \end{aligned} \tag{33}$$

The controller is

$$
\mu = \begin{bmatrix} k\_1 & k\_2 \end{bmatrix} \begin{bmatrix} e\_{\ge 1} \\ e\_{\ge 2} \end{bmatrix} = k\_1 e + k\_2 \begin{pmatrix} \mathbf{x}\_{2\pi} \cdot \mathbf{x}\_2 \end{pmatrix} \tag{34}
$$

#### **6.1.1 Parallel controller**

$$
\mu = k\_1 \left(\mathbf{y}\_r \cdot \mathbf{y}\right) + k\_2 \left(\mathbf{x}\_{2r} \cdot \mathbf{x}\_2\right) \tag{35}
$$

#### **6.1.2 Cascade controller**

$$
\mu = k\_2 \left[ (\mathbf{x}\_{2r} \cdot \mathbf{x}\_2) + \frac{k\_1}{k\_2} (\mathbf{y}\_r \cdot \mathbf{y}) \right] \tag{36}
$$

#### **6.1.3 One block controller**

To get the one block controller we substitute (30) and get (in Laplace domain)

$$\begin{aligned} \mu(s) &= k\_1 e + k\_2 \left( \mathbf{x}\_{2r} \cdot \mathbf{x}\_2 \right) = k\_1 e + \frac{k\_2}{s} \frac{\mathbf{x}\_2}{\mathbf{x}\_1} \left( \mathbf{x}\_{1r} \cdot \mathbf{x}\_1 \right) \\ \mathcal{L}(s) &= \frac{\mu(s)}{c(s)} = k\_1 + k\_2 \frac{b\_2 s - a\_2 b}{b\_1 s + (b\_2 + a\_1 b\_1)} = k\_P + \frac{k\_D s}{s \tau\_D + 1} \end{aligned} \tag{37}$$

Family of the PID Controllers 41

( ) x -x y -y *ks k us k*

To get the one block output controller derive we substitute (30) and get (in Laplace domain)

1 2 2r 2 1 1r 1

*k k k x u s = k e k e ke <sup>e</sup>*

( ) x -x x -x

1 2 11

*u s bs ab <sup>k</sup> k ks Cs = k k <sup>k</sup> es bs b ab s s s*

1 2

stable proper PID controller, i.e. no direct derivative is required.

<sup>1</sup> , *1r 1 x1 2r 2 x2 x -x e x -x e* 

> *x1 x2*

 

*e e*

> *x1 x2*

 

*e dt*

1. Remarks in section 6.1.4 apply here mutatis mutandis.

**6.3 PID controller in PIV configuration** 

 1 3 2 2r 2 r

3 3 2 2

2 2 3

( ) ( ) ( ) ( ) <sup>1</sup>

2. For 2nd order plant with a stable zero, the optimal controller with one integrator is a

Zero steady state tracking error on the output and the second state (velocity) is required.

1 = ,

<sup>1234</sup> 123 4

*ukkkk k e k e k e dt k e dt e dt*

1 x1 2 x2 x1 x2 () e e e e *<sup>k</sup> <sup>k</sup> u s = k k*

<sup>1</sup> r 2 2r 2 ( ) y -y x -x

*s s*

*<sup>k</sup> <sup>k</sup> u s = k <sup>k</sup>*

3 4

*s s*

<sup>3</sup> <sup>4</sup>

1

*s sx s*

*P*

*x1 x2*

 

*e dt e dt*

*x1 x2 x1 x2*

(47)

2

*k s*

(42)

*I D*

*D*

(44)

(43)

(45)

(46)

**6.2.2 Cascade controller** 

**6.2.3 One block controller** 

This is the PID controller.

**6.2.4 Discussion** 

The tracking errors are

and in Laplace domain

**6.3.1 Parallel controller** 

The controller is

This is the PD controller.

#### **6.1.4 Discussion**


#### **6.2 PID controller**

Zero steady state tracking error on the output is required. The tracking errors are

$$\begin{aligned} e\_{x1} &= \mathbf{y}\_{\mathbf{r}} \cdot \mathbf{y}\_{\mathbf{r}} &= \mathbf{x}\_{1\mathbf{r}} \cdot \mathbf{x}\_1 = e \\ e\_{x2} &= \mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2 \\ \dot{\eta}\_1 &= e\_{x1} \end{aligned} \tag{38}$$

The controller is

$$\mathbf{u} = \begin{bmatrix} k\_1 & k\_2 & k\_3 \end{bmatrix} \begin{bmatrix} e\_{x1} \\ e\_{x2} \\ \eta\_1 \end{bmatrix} = k\_1 e + k\_2 \left( \mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2 \right) + k\_3 \int e dt \tag{39}$$

and the controller in Laplace domain

$$
\mu(\mathbf{s}) = k\_1 e + k\_2 \left( \mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2 \right) + \frac{k\_3}{s} e
\tag{40}
$$

#### **6.2.1 Parallel controller**

$$
\mu = \left(k\_1 + \frac{k\_3}{s}\right) \left(\mathbf{y}\_\mathbf{r} \cdot \mathbf{y}\right) + k\_2 \left(\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) \tag{41}
$$

1. We used the assumption that 21 2 1 ( )/ ( ) ( )/ ( ) *r r xs xs x s x s* and ignored the response to

2. For 2nd order plant with a stable zero the optimal controller is a proper PD controller,

3. The pole/filter of the derivative in (37) cancels out the zero of the plant (28). This is optimal/correct for deterministic (noiseless) systems. For systems with noisy

4. The cancellation of the plant's zero by the optimal controller creates an uncontrollable system. This may work (although is not good practice) for stable zero. However, when the plant has non-minimum phase (unstable) zero the optimal PD controller induces uncontrollable unstable mode, which means that the Optimal PD controller

5. As for a plant with unstable zero the optimal one block PID controller cannot be realized, then measurement of the two states, or an observer is required if one wishes to

6. If stable controller is required it is possible to implement the optimal PD one block

7. For 2nd order system without zero the deterministic optimal controller is not proper, i.e.

1 r r 1r 1

y -y x -x

*e e*

cannot/should not be implemented in the one block controller architecture.

Zero steady state tracking error on the output is required. The tracking errors are

*x x*

*e e*

2 2r 2 1 1

*x*

 1

1 u= x -x *x x e*

> *<sup>k</sup> u k <sup>k</sup> s*

 

1 2 3 2 1 2 2r 2 3

1 2 2r 2 ( ) x -x *<sup>k</sup> u s = k e k <sup>e</sup>*

*k k k e k e k k edt*

<sup>3</sup>

 <sup>3</sup> <sup>1</sup> yr 2 2r 2 -y x -x

*s*

(40)

(41)

 x -x =

 

measurements this cancelation is no more optimal (Rusnak, 2000b).

architecture controller only for minimum phase plants!

This is the PD controller.

initial conditions.

i.e. no direct derivative is required.

build the optimal controller.

requires pure derivative.

and the controller in Laplace domain

**6.2.1 Parallel controller** 

**6.2 PID controller** 

The controller is

**6.1.4 Discussion** 

#### **6.2.2 Cascade controller**

$$\ln(\mathbf{s}) = k\_2 \left[ (\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2) + \frac{k\_1 \mathbf{s} + k\_3}{k\_2 \mathbf{s}} (\mathbf{y}\_\mathbf{r} \cdot \mathbf{y}) \right] \tag{42}$$

#### **6.2.3 One block controller**

To get the one block output controller derive we substitute (30) and get (in Laplace domain)

$$\begin{aligned} \mu(s) &= k\_1 e + k\_2 \left( \mathbf{x}\_{2r} \cdot \mathbf{x}\_2 \right) + \frac{k\_3}{s} e = k\_1 e + \frac{k\_2}{s} \frac{\mathbf{x}\_2}{\mathbf{x}\_1} \left( \mathbf{x}\_{1r} \cdot \mathbf{x}\_1 \right) + \frac{k\_3}{s} e \\ \mathbf{C}(s) &= \frac{\mu(s)}{c(s)} = k\_1 + k\_2 \frac{b\_2 s - a\_2 b}{b\_1 s + (b\_2 + a\_1 b\_1)} + \frac{k\_3}{s} = k\_P + \frac{k\_I}{s} + \frac{k\_D s}{s \mathbf{r}\_D + 1} \end{aligned} \tag{43}$$

This is the PID controller.

#### **6.2.4 Discussion**


#### **6.3 PID controller in PIV configuration**

Zero steady state tracking error on the output and the second state (velocity) is required. The tracking errors are

$$
\boldsymbol{\eta}\_{1} = \begin{bmatrix} \mathbf{x}\_{1r} \ \mathbf{-} \mathbf{x}\_{1} \\ \mathbf{x}\_{2r} \ \mathbf{-} \mathbf{x}\_{2} \end{bmatrix} = \begin{bmatrix} \mathbf{e}\_{\times 1} \\ \mathbf{e}\_{\times 2} \end{bmatrix} \prime \qquad \boldsymbol{\eta}\_{1} = \begin{bmatrix} \int \mathbf{e}\_{\times 1} dt \\ \int \mathbf{e}\_{\times 2} dt \end{bmatrix} \prime \tag{44}
$$

The controller is

(38)

(39)

$$u = \begin{bmatrix} k\_1 & k\_2 & k\_3 & k\_4 \end{bmatrix} \begin{bmatrix} e\_{x1} \\ e\_{x2} \\ \int e\_{x1} dt \\ \int e\_{x1} dt \\ \int e\_{x2} dt \end{bmatrix} = k\_1 e\_{x1} + k\_2 e\_{x2} + k\_3 \int e\_{x1} dt + k\_4 \int e\_{x2} dt \tag{45}$$

and in Laplace domain

$$\mathbf{u}(\mathbf{s}) \equiv \mathbf{k}\_1 \mathbf{e}\_{\times 1} + \mathbf{k}\_2 \mathbf{e}\_{\times 2} + \frac{\mathbf{k}\_3}{\mathbf{s}} \mathbf{e}\_{\times 1} + \frac{\mathbf{k}\_4}{\mathbf{s}} \mathbf{e}\_{\times 2} \tag{46}$$

#### **6.3.1 Parallel controller**

$$
\mu(\mathbf{x}) = \left(k\_1 + \frac{k\_3}{s}\right) \mathbf{(y}\_\mathbf{r} \cdot \mathbf{y}) + \left(k\_2 + \frac{k\_4}{s}\right) \mathbf{(x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) \tag{47}
$$

Family of the PID Controllers 43

1 2 2r 2 <sup>2</sup> ( ) x -x *<sup>k</sup> <sup>k</sup> u s = k e k e e*

*<sup>k</sup> <sup>k</sup> u k <sup>k</sup> s s*

*us k*

**6.4.1 Parallel controller** 

**6.4.2 Cascade controller** 

**6.4.3 One block output controller** 

This is the PI2D controller.

in different configuration, i.e.

1 , *1r 1 x1 2r 2 x2 x -x e x -x e* 

**6.4.4 Discussion** 

**6.5 PI<sup>2</sup>**

The controller is

<sup>3</sup> <sup>4</sup>

 <sup>3</sup> <sup>4</sup> <sup>1</sup> <sup>2</sup> yr 2 2r 2 -y x -x

 2 1 34

*ks ks k*

*k s* 

 3 3 <sup>4</sup> 4 2 1 2 2 1r 1 2 2r 2 1 1r 1 2 1r 1

*P*

 2 1 21 10 , *x1* 

> <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup>

*<sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>x</sup>*

*<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup>*

2 1

 

> 

*x x*

*e dt*

 *or e e* (56)

2 2r 2 2 r 2

2 2 3 4 2

Here we want to force zero steady state tracking error on the second state, as well, however

 <sup>1</sup> = , *2r 2*

11 22 3 1 4 1

*xx x x*

*k e k e k e dt k e dt*

1234 1234 <sup>1</sup> <sup>1</sup>

*ukkkk kkkk e dt*

*e x -x*

 

( ) x -x x -x x -x x -x

*k k k k x*

*s s s s x*

( ) x -x y -y

Two integrators in the position loop and proportional feedback in the velocity loop.

1 2 2 2

( ) ( ) ( ) ( ) <sup>1</sup>

*u s bs ab k k k k s <sup>k</sup> Cs k k <sup>k</sup> es bs b ab s s s s s*

*u s k kk k*

To get the one block controller we substitute (30) and get (in Laplace domain)

1 2 11

1. Remarks in section 6.1.4 apply here mutatis mutandis.

**D controller in IPIV configuration** 

*s s*

(52)

(54)

(55)

(57)

1

(53)

*II D*

*D*

#### **6.3.2 Cascade controller - the PIV configuration**

$$
\mu(\mathbf{s}) = \left(k\_2 + \frac{k\_4}{s}\right) \left[ (\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2) + \frac{k\_1 \mathbf{s} + k\_3}{k\_2 \mathbf{s} + k\_4} (\mathbf{y}\_\mathbf{r} \cdot \mathbf{y}) \right] \tag{48}
$$

This is called the PIV configuration (Proportional feedback in position loop and proportional+integral feedback in the velocity loop) (configuration=combination of architecture and structure) as there is almost proportional feedback (Lead-Lag) on the position <sup>1</sup> x and then in the velocity loop on 2 x there is proportional and one integral feedback.

#### **6.3.3 One block output controller**

To get the one block controller we substitute (30) and get (in Laplace domain)

$$\begin{aligned} \mu(s) &= \left(k\_1 + \frac{k\_3}{s}\right)e\_{\ge 1} + \left(k\_2 + \frac{k\_4}{s}\right)e\_{\ge 2} \\ C(s) &= \frac{\mu(s)}{c(s)} = \left(k\_1 + \frac{k\_3}{s}\right) + \left(k\_2 + \frac{k\_4}{s}\right)\frac{\chi\_2}{\chi\_1} = k\_P + \frac{k\_I}{s} + \frac{k\_D}{s\tau\_D + 1} \end{aligned} \tag{49}$$

This is the PID controller.

#### **6.3.4 Discussion**


#### **6.4 PI<sup>2</sup> D controller**

Zero steady state tracking error on the output for ramp input or disturbance is required. The tracking errors are

$$\begin{aligned} \dot{\eta}\_1 &= e\_{x1} = y\_r \ \cdot \ y = \mathbf{x}\_{1r} \ \cdot \ \mathbf{x}\_1 = e \\\ e\_{x2} &= \mathbf{x}\_{2r} \ \cdot \ \mathbf{x}\_2 \\\ \dot{\eta}\_2 &= \eta\_1 \end{aligned} \tag{50}$$

The controller is

$$\begin{aligned} \boldsymbol{\mu} &= \begin{bmatrix} k\_1 & k\_2 & k\_3 & k\_4 \end{bmatrix} \begin{bmatrix} \mathbf{e}\_{\boldsymbol{x}\boldsymbol{t}} \\ \mathbf{e}\_{\boldsymbol{x}\boldsymbol{t}} \\ \eta\_{\boldsymbol{t}} \\ \eta\_{\boldsymbol{2}} \end{bmatrix} = \begin{bmatrix} k\_1 & k\_2 & k\_3 & k\_4 \end{bmatrix} \begin{bmatrix} \mathbf{e}\_{\boldsymbol{x}\boldsymbol{t}} \\ \mathbf{e}\_{\boldsymbol{x}\boldsymbol{2}} \\ \end{bmatrix} \\ &= k\_1 \mathbf{e}\_{\boldsymbol{x}\boldsymbol{1}} + k\_2 \mathbf{e}\_{\boldsymbol{x}\boldsymbol{2}} + k\_3 \int \mathbf{e}\_{\boldsymbol{x}\boldsymbol{1}} dt + k\_4 \iint \mathbf{e}\_{\boldsymbol{x}\boldsymbol{t}} dt \end{aligned} \tag{51}$$

and in Laplace domain

( ) x -x y -y *<sup>k</sup> ks k u s = k*

This is called the PIV configuration (Proportional feedback in position loop and proportional+integral feedback in the velocity loop) (configuration=combination of architecture and structure) as there is almost proportional feedback (Lead-Lag) on the position

3 4 2

4. Although formally the cascade architecture controller requires the tuning of six parameters in (48), the deterministic optimal PIV controller needs the tuning of four

Zero steady state tracking error on the output for ramp input or disturbance is required. The

*e y y x xe*

1 1 11

*xr r*


1234 1234

x 1

e

e

x 2

1

η

η

2

*ukkkk kkkk*

2 22


123 4 *x1 x2 x1 x1*

*k e k e k e dt k e dt*

 

*x r*

2 1

 

*x1 x2*

*u s <sup>k</sup> kx k k Cs = k k k es s s x s s*

( ) ( ) ( ) <sup>1</sup>

1

*P*

*e xx* (50)

  x 1

e

e

e d t

x 2

x 1

> x 1

e d t

*I D*

*D*

<sup>1</sup> x and then in the velocity loop on 2 x there is proportional and one integral feedback.

To get the one block controller we substitute (30) and get (in Laplace domain)

3 4 1 2

*s s*

*<sup>k</sup> <sup>k</sup> us k e k e*

1. Remarks in section 6.1.4 apply here mutatis mutandis.

parameters only, as can be deduced from (46).

1 2

2. Two different tracking problems (38, 44) lead to the same one block controller. 3. In the parallel architecture there is a PI controller in each of the errors (47).

*s ks k* 

 <sup>4</sup> 1 3 2 2r 2 r

2 4

(48)

(49)

(51)

**6.3.2 Cascade controller - the PIV configuration** 

**6.3.3 One block output controller** 

This is the PID controller.

**D controller** 

tracking errors are

The controller is

and in Laplace domain

**6.3.4 Discussion** 

**6.4 PI<sup>2</sup>**

( )

$$
\mu(\mathbf{s}) = k\_1 e + k\_2 \left( \mathbf{x}\_{2r} \cdot \mathbf{x}\_2 \right) + \frac{k\_3}{s} e + \frac{k\_4}{s^2} e \tag{52}
$$

## **6.4.1 Parallel controller**

$$
\mu = \left(k\_1 + \frac{k\_3}{s} + \frac{k\_4}{s^2}\right) \left(\mathbf{y}\_\mathbf{r} \cdot \mathbf{y}\right) + k\_2 \left(\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) \tag{53}
$$

#### **6.4.2 Cascade controller**

$$\mathbf{u}(\mathbf{s}) = k\_2 \left[ \left( \mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2 \right) + \frac{\left( k\_1 \mathbf{s}^2 + k\_3 \mathbf{s} + k\_4 \right)}{k\_2 \mathbf{s}^2} \left( \mathbf{y}\_\mathbf{r} \cdot \mathbf{y} \right) \right] \tag{54}$$

Two integrators in the position loop and proportional feedback in the velocity loop.

#### **6.4.3 One block output controller**

To get the one block controller we substitute (30) and get (in Laplace domain)

$$\begin{aligned} u(s) &= \left(k\_1 + \frac{k\_3}{s} + \frac{k\_4}{s^2}\right) (\mathbf{x}\_{1\tau} \cdot \mathbf{x}\_1) + k\_2 \left(\mathbf{x}\_{2\tau} \cdot \mathbf{x}\_2\right) = \left(k\_1 + \frac{k\_3}{s} + \frac{k\_4}{s^2}\right) (\mathbf{x}\_{1\tau} \cdot \mathbf{x}\_1) + k\_2 \frac{\mathbf{x}\_2}{\mathbf{x}\_1} (\mathbf{x}\_{1\tau} \cdot \mathbf{x}\_1) \\ C(s) &= \frac{u(s)}{e(s)} = k\_1 + k\_2 \frac{b\_2 s - a\_2 b}{b\_1 s + (b\_2 + a\_1 b\_1)} + \frac{k\_3}{s} + \frac{k\_4}{s^2} = k\_P + \frac{k\_1}{s} + \frac{k\_{12}}{s^2} + \frac{k\_D s}{s\tau\_D + 1} \end{aligned} \tag{55}$$

This is the PI2D controller.

#### **6.4.4 Discussion**

1. Remarks in section 6.1.4 apply here mutatis mutandis.

#### **6.5 PI<sup>2</sup> D controller in IPIV configuration**

Here we want to force zero steady state tracking error on the second state, as well, however in different configuration, i.e.

$$
\dot{\eta}\_1 = \begin{bmatrix} \mathbf{x}\_{1r} \cdot \mathbf{x}\_1 \\ \mathbf{x}\_{2r} \cdot \mathbf{x}\_2 \end{bmatrix} = \begin{bmatrix} e\_{\mathbf{x}1} \\ e\_{\mathbf{x}2} \end{bmatrix}, \qquad \eta\_1 = \begin{bmatrix} e & \\ \int \begin{bmatrix} e & \\ \end{bmatrix} \end{bmatrix}, \qquad \dot{\eta}\_2 = \begin{bmatrix} 1 & 0 \end{bmatrix}
\eta\_1, \text{ or } \ddot{\eta}\_2 = \dot{\eta}\_1 = e\_{\mathbf{x}1} = e \tag{56}
$$

The controller is

$$\begin{aligned} \mathbf{u} &= \begin{bmatrix} k\_1 & k\_2 & k\_3 & k\_4 \end{bmatrix} \begin{bmatrix} e\_{x1} \\ e\_{x2} \\ \eta\_1 \\ \eta\_2 \end{bmatrix} = \begin{bmatrix} k\_1 & k\_2 & k\_3 & k\_4 \end{bmatrix} \begin{bmatrix} e\_{x1} \\ e\_{x2} \\ \int e\_{x1} dt \\ \iint e\_{x1} dt \end{bmatrix} \\ &= k\_1 e\_{x1} + k\_2 e\_{x2} + k\_3 \int e\_{x1} dt + k\_4 \iint e\_{x1} dt \end{aligned} \tag{57}$$

Family of the PID Controllers 45

1 2 3 4 5 6

1 2 2r 2 2r 2 2 2 2r 2 ( ) x -x x -x x -x

*s s s s* 

**V configuration** 

*ks ks k ks ks k*

*<sup>k</sup> <sup>k</sup> k k us ke k <sup>e</sup> <sup>e</sup>*

*k k <sup>k</sup> <sup>k</sup> u k <sup>k</sup>*

2 2 2 46 1 35

*u k k k k k k*

and in Laplace domain

**6.6.1 Parallel controller** 

**6.6.2 Cascade controller – the PI<sup>2</sup>**

*u*

proportional and two integrals feedback.

**6.6.3 One block output controller** 

This is the PI2D controller.

**6.6.4 Discussion** 

**6.7 Summary** 

 

*dt dt dt dt*

*x2 x1 x2 x1 x2 x1*

*e e e e e e*

 

<sup>3</sup> <sup>4</sup> 5 6

 3 5 <sup>4</sup> <sup>6</sup> <sup>1</sup> 2 2 yr 2 2r 2 -y x -x

2 2 2r 2 r

 3 5 <sup>4</sup> <sup>6</sup> 1 2 2 r 2 2r 2

1 2 2 2

1

This section presented the family of the generalized PID controllers for 2nd order systems. The following tables summarize the structure of the controllers in the different architectures.

*D*

( ) ( ) ( ) ( )

*es s s s s bs b ab*

( ) y -y x -x

*s s s s u s kk k k bs ab Cs = = k <sup>k</sup>*

> 1 2 2

1. Remarks in section 6.1.4 apply here mutatis mutandis.

*k k ks <sup>k</sup> s s s*

*P*

*II D*

*k k <sup>k</sup> <sup>k</sup> us k <sup>k</sup>*

3 5 4 22 6

1 2 11

*s ks ks k* 

This is called the PI2V configuration (Proportional feedback in position loop and proportional +double integral feedback in the velocity loop) as there is almost proportional feedback (Lead-Lag) in the position loop, on *y,* and then in the velocity loop, on *x2*, there is

2 46 x -x y -y

*s s s s* (64)

 

(63)

(65)

(66)

(67)

and in Laplace domain

$$
\mu(\mathbf{s}) = k\_1 e + k\_2 \left( \mathbf{x}\_{2r} \cdot \mathbf{x}\_2 \right) + \frac{k\_3}{s} e + \frac{k\_4}{s} \left( \mathbf{x}\_{2r} \cdot \mathbf{x}\_2 \right) + \frac{k\_5}{s^2} e \tag{58}
$$

#### **6.5.1 Parallel implementation**

$$
\mu = \left(k\_1 + \frac{k\_3}{s} + \frac{k\_5}{s^2}\right) \left(y\_r \cdot y\right) + \left(\begin{array}{c} k\_2 \ \leftarrow \frac{k\_1}{s} \end{array}\right) \left(\mathbf{x}\_{2r} \cdot \mathbf{x}\_2\right) \tag{59}
$$

#### **6.5.2 Cascade controller – the IPIV configuration**

$$\mu(\mathbf{s}) = \frac{k\_2 \mathbf{s} + k\_4}{s} \left[ \left( \mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2 \right) + \frac{k\_1 \mathbf{s}^2 + k\_3 \mathbf{s} + k\_5}{s \left( k\_2 \mathbf{s} + k\_4 \right)} \left( \mathbf{y}\_\mathbf{r} \cdot \mathbf{y} \right) \right] \tag{60}$$

This is called the IPIV configuration (Proportional +integral feedback in position loop and proportional +integral feedback in the velocity loop) as there is almost proportional feedback on the position loop, *y*, and then in the velocity loop on, *x2*, there is proportional and one integral feedback.

#### **6.5.3 One block output controller**

To get the one block output controller we substitute (30) and get (in Laplace domain)

$$\begin{aligned} u(s) &= k\_1 e + k\_2 \left( \mathbf{x}\_{2\tau} \cdot \mathbf{x}\_2 \right) + \frac{k\_3}{s} e + \frac{k\_5}{s^2} e + \frac{k\_4}{s} \left( \mathbf{x}\_{2\tau} \cdot \mathbf{x}\_2 \right) \\ C(s) &= \frac{u(s)}{c(s)} = k\_1 + k\_2 \frac{\mathbf{x}\_2}{\mathbf{x}\_1} + \frac{k\_3}{s} + \frac{k\_5}{s^2} + \frac{k\_4}{s} \frac{\mathbf{x}\_2}{\mathbf{x}\_1} = k\_P + \frac{k\_{11}}{s} + \frac{k\_{12}}{s^2} + \frac{k\_{D^S}}{s \tau\_D + 1} \end{aligned} \tag{61}$$

This is the PI2D controller.

#### **6.5.4 Discussion**

1. Remarks in section 6.1.4 apply here mutatis mutandis.

#### **6.6 PI<sup>2</sup> D controller in PI<sup>2</sup> V configuration**

Here we want to force zero steady state tracking error on the rate of the output as well, and

$$
\dot{\eta}\_1 = \begin{bmatrix} \mathbf{x}\_1 \cdot \mathbf{x}\_{1\mathbf{r}} \\ \mathbf{x}\_2 \cdot \mathbf{x}\_{2\mathbf{r}} \end{bmatrix} = \begin{bmatrix} e\_{\times 1} \\ e\_{\times 2} \end{bmatrix}' \qquad \qquad \eta\_1 = \begin{bmatrix} \int e\_{\times 1} dt \\ \int e\_{\times 2} dt \end{bmatrix}' \tag{62}
$$

$$
\dot{\eta}\_2 = \eta\_1 \tag{62}
$$

The controller is

1 2 2r 2 2r 2 <sup>2</sup> ( ) x -x x -x *k k <sup>k</sup> u s = k e k <sup>e</sup> <sup>e</sup>*

2 4 1 35

( ) x -x y -y *ks k ks ks k*

This is called the IPIV configuration (Proportional +integral feedback in position loop and proportional +integral feedback in the velocity loop) as there is almost proportional feedback on the position loop, *y*, and then in the velocity loop on, *x2*, there is proportional

*s sks k*

To get the one block output controller we substitute (30) and get (in Laplace domain)

 3 5 <sup>4</sup> 1 2 2r 2 2 2r 2

*s s s us x k k k x k k ks C s = k k <sup>k</sup> es x s s x s s s s*

( ) ( ) ( ) <sup>1</sup>

Here we want to force zero steady state tracking error on the rate of the output as well,

x1

2 1 

( ) x -x x -x

*k k <sup>k</sup> u s = k e k e e*

1. Remarks in section 6.1.4 apply here mutatis mutandis.

1

**V configuration** 

1 1r

 

2 2r x -x , x -x *x1 x2 e e*

*s s*

**6.5.2 Cascade controller – the IPIV configuration** 

*u s*

3 5 <sup>4</sup>

 3 5 1 2 *r 2r 2 k k u k <sup>y</sup> - <sup>y</sup> x -x*

k+

2

 2

2r 2 r 2 4

2 3 5 42 1 2 1 2 2 2 1 1

*P*

<sup>1</sup> = , *x2*

*e dt e dt*

(62)

 

*s s s*

k

s4

(60)

*II D*

*D*

(58)

(59)

(61)

and in Laplace domain

**6.5.1 Parallel implementation** 

and one integral feedback.

This is the PI2D controller.

**D controller in PI<sup>2</sup>**

**6.5.4 Discussion** 

**6.6 PI<sup>2</sup>**

and

The controller is

**6.5.3 One block output controller** 

$$\mu = \begin{bmatrix} \mathbf{k}\_1 & \mathbf{k}\_2 & \mathbf{k}\_3 & \mathbf{k}\_4 & \mathbf{k}\_5 & \mathbf{k}\_6 \end{bmatrix} \begin{bmatrix} \mathbf{e}\_{xl} \\ \mathbf{e}\_{xl} \\ \begin{bmatrix} \mathbf{e}\_{xl} dt \\ \mathbf{e}\_{x2} dt \\ \begin{bmatrix} \mathbf{e}\_{x2} dt \\ \mathbf{d} \end{bmatrix} \\ \begin{bmatrix} \mathbf{e}\_{xl} dt \\ \mathbf{d} \end{bmatrix} \end{bmatrix} \tag{63}$$

and in Laplace domain

$$\ln(\mathbf{s}) = k\_1 \mathbf{e} + k\_2 \left(\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) + \frac{k\_3}{\mathbf{s}} \mathbf{e} + \frac{k\_4}{\mathbf{s}} \left(\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) + \frac{k\_5}{\mathbf{s}^2} \mathbf{e} + \frac{k\_6}{\mathbf{s}^2} \left(\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) \tag{64}$$

#### **6.6.1 Parallel controller**

$$
\mu = \left(k\_1 + \frac{k\_3}{s} + \frac{k\_5}{s^2}\right) \left(\mathbf{y}\_\mathbf{r} \cdot \mathbf{y}\right) + \left(k\_2 + \frac{k\_4}{s} + \frac{k\_6}{s^2}\right) \left(\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2\right) \tag{65}
$$

#### **6.6.2 Cascade controller – the PI<sup>2</sup> V configuration**

$$\ln u = \frac{k\_2 s^2 + k\_4 s + k\_6}{s^2} \left[ (\mathbf{x}\_{2\mathbf{r}} \cdot \mathbf{x}\_2) + \frac{k\_1 s^2 + k\_3 s + k\_5}{k\_2 s^2 + k\_4 s + k\_6} (\mathbf{y}\_{\mathbf{r}} \cdot \mathbf{y}) \right] \tag{66}$$

This is called the PI2V configuration (Proportional feedback in position loop and proportional +double integral feedback in the velocity loop) as there is almost proportional feedback (Lead-Lag) in the position loop, on *y,* and then in the velocity loop, on *x2*, there is proportional and two integrals feedback.

#### **6.6.3 One block output controller**

$$\begin{split} u(s) &= \left(k\_1 + \frac{k\_3}{s} + \frac{k\_5}{s^2}\right) \mathbf{(y\_\tau \cdot \mathbf{y})} + \left(k\_2 + \frac{k\_4}{s} + \frac{k\_6}{s^2}\right) \mathbf{(x\_{2\tau} - \mathbf{x\_2})}\\ C(s) &= \frac{u(s)}{c(s)} = \left(k\_1 + \frac{k\_3}{s} + \frac{k\_5}{s^2}\right) + \left(k\_2 + \frac{k\_4}{s} + \frac{k\_6}{s^2}\right) \frac{b\_2s - a\_2b}{b\_1s + (b\_2 + a\_1b\_1)}\\ &= k\_P + \frac{k\_{I1}}{s} + \frac{k\_{I2}}{s^2} + \frac{k\_Ds}{s\tau\_D + 1} \end{split} \tag{67}$$

This is the PI2D controller.

#### **6.6.4 Discussion**

1. Remarks in section 6.1.4 apply here mutatis mutandis.

#### **6.7 Summary**

This section presented the family of the generalized PID controllers for 2nd order systems. The following tables summarize the structure of the controllers in the different architectures.

Family of the PID Controllers 47

The reference trajectory generator encapsulates the required closed loop behavior as stated by the system specification-requirements. There can be two cases: the trajectory is either unknown or known in advance. The former case gives the well known pre-filter that creates the feed-forward as well. In the second case, for example, minimum time trajectories for limited acceleration or jerk, minimum acceleration or jerk energy trajectories, or any other profile can be required. Both cases are presented in (Leonhard, 1996, pp. 80, 347, 363-364,

In this chapter the generalized PID controllers for 1st and 2nd order system that are able to drive the tracking error to zero for up to second order polynomials inputs and disturbances have been derived. This presented in detail a methodology to derive additional members of the family of generalized PID controllers for high order system (Rusnak, 1999) and high

Following the theory and the author's experience the full state feedback, especially the cascade architecture, Figure 2, is preferable over the one block controller, Figure 3. This may come at the expense of higher cost. However in modern digital control loop that are using absolute or incremental encoders the position and velocity information is derived at the

The motion control engineers prefer the cascade controller because of implementation and tuning easiness. The most important feature is that in the cascade architecture the feedback loop can be tuned sequentially. That is, start with the velocity-inner loop, that is usually high bandwidth, and then to proceed to the position-outer loop. The same apply to higher

By the use of LQR theory we formulated a control-tracking problem and showed those cases when their solution gives members of the PImDm-1 family of controllers. This way heuristics are avoided and a systematic approach to explanation for the excellent performance of the PID controllers is given. The well known one block PID controller architecture is optimal for

Linear Quadratic Tracking problem of 2nd order systems with no zero or stable zero.

order polynomial inputs and disturbances. These are the PImDn-1 controllers.

Controller type plant integral action(m) § PD kP+kD s no zero 0 6.1 PD kP+kD s/(s D+1) zero 0 6.1 PID kP + kI/s+kDs no zero 1 6.2,3 PID kP+ kI/s+kD s/(sD+1) zero 1 6.2,3 PI2D kP +kI1/s+kI2/s2+kDs no zero 2 6.4,5,6 PI2D kP+kI/s+kI1/s+kI2/s2+kDs/(s D+1) zero 2 6.4,5,6 Table 5. The structure of one block generalized PID controller for 2nd order plant with and

One block PD, PID and generalized PID controller (Figure 3)

without minimum phase zero.

**7. Reference trajectory generator** 

367) and in many other publications.

order generalized PID controllers.

**8. Discussion** 

same cost.

**9. Conclusions** 

Table 2 presents the family of generalized PID controllers for 2nd order systems in the parallel architecture that are able to drive the tracking error to zero for up to constant acceleration input and disturbance. Formally, if all possible parallel configurations are enumerated then there are three more parallel structures as detailed in Table 3. However these additional structures are equivalent to the respective structures in Table 2 as detailed in the rightmost column. Therefore these configurations are not considered in the following.


Table 2. The structure of the parallel architecture controllers for 2nd order plant.


Table 3. The structure of the parallel architecture controllers for 2nd order plant.

Tables 4 and 5 present the family of generalized PID controllers for 2nd order systems in the cascade architecture and in the one block controller architecture, respectively, that are able to drive the tracking error to zero for up to constant acceleration input and disturbance.


Table 4. The structure of the cascade architecture controllers for 2nd order plant.

Table 2 presents the family of generalized PID controllers for 2nd order systems in the parallel architecture that are able to drive the tracking error to zero for up to constant acceleration input and disturbance. Formally, if all possible parallel configurations are enumerated then there are three more parallel structures as detailed in Table 3. However these additional structures are equivalent to the respective structures in Table 2 as detailed in the rightmost column. Therefore these configurations are not considered in the

*Cx1 Cx2* §

*Cx1 Cx2* §

*Cp* (position-outer loop) *Cv* (velocity-inner loop) §

PID k1 k2+k4/s 6.2 PI2D k1 k2+k4/s+k6/s2 6.4 PI2D-IPIV k1+k3/s k2+k4/s+k6/s2 6.5

Tables 4 and 5 present the family of generalized PID controllers for 2nd order systems in the cascade architecture and in the one block controller architecture, respectively, that are able to drive the tracking error to zero for up to constant acceleration input and

PD k1 k1/k2 6.1 PID (k1s+k3)/k2/s k2 6.2 PIV (k1s+k3)/(k2s+k4) (k2s+k4)/s 6.3 PI2D (k1s2+k3s+k4)/k2/s2 k2 6.4 IPIV (k1s2+k3s+k5)/s(k2s+k4) (k2s+k4)/s 6.5 PI2V (k1s2+k3s+k5)/(k2s2+k4s+k6) (k2s2+k2s+k6)/s2 6.6

Table 4. The structure of the cascade architecture controllers for 2nd order plant.

PD k1 k2 6.1 PID k1+k3/s k2 6.2 PID- PIV k1+k3/s k2+k4/s 6.3 PI2D k1+k3/s+k4/s2 k2 6.4 PI2D-IPIV k1+k3/s+k5/s2 k2+k4/s 6.5 PI2D- PI2V k1+k3/s+k5/s2 k2+k4/s+k6/s2 6.6

Table 2. The structure of the parallel architecture controllers for 2nd order plant.

Table 3. The structure of the parallel architecture controllers for 2nd order plant.

Generalized PID controller - Parallel architecture (Figure 1)

Generalized PID controller - Parallel architecture (Figure 1)

Generalized PID controller - Cascade architecture (Figure 2)

following.

disturbance.


Table 5. The structure of one block generalized PID controller for 2nd order plant with and without minimum phase zero.
