**2.1.2 Principles of controller structure selection based on the plant type**

Industrial process variables (e.g. position, speed, current, temperature, pressure, humidity, level etc.) are commonly controlled using PI controllers. In practice, the derivative part is usually switched off due to measurement noise. For pressure and level control in gas tanks, using P controller is sufficient (Bakošová & Fikar, 2008). However, adding derivative part improves closed-loop stability and steepens the step response rise (Balátě, 2004).

Time response of the controlled variable y(t) is modifiable by tuning proportional gain K, and integrating and derivative time constants Ti and Td, respectively; the objective is to achieve a zero steady-state control error e(t) irrespective if caused by changes in the reference w(t) or the disturbance d(t). This section presents practice-oriented PID controller design methods based on various perfomance criteria. Consider the control-loop in Fig. 1

n(t)

<sup>1</sup>w(t) e(t) u(t) y(t)

SW <sup>3</sup>

d(t)

G(s)

2

A controller design is a two-step procedure consisting of controller structure selection (P, PI,

Appropriate structure of the controller GR(s) is usually selected with respect to zero steady-

Consider the feedback control loop in Fig. 1 where G(s) is the plant transfer function.

*<sup>q</sup> s s <sup>s</sup> <sup>L</sup> <sup>s</sup> e sE s s W s q w L s s K*

*q*

(1)

*q* (2)

Fig. 1. Feedback control-loop with load disturbance d(t) and measurement noise n(t)

**2.1.1 Controller structure selection based on zero steady-state error condition** 

 0 0 <sup>0</sup> <sup>1</sup> lim ( ) lim ( ) ! lim 1 ()

 

is zero if in the open-loop L(s)=G(s)GR(s), the integrator degree L=S+R is greater than the

*L* 

where S and R are integrator degrees of the plant and controller, respectively, KL is open-

Industrial process variables (e.g. position, speed, current, temperature, pressure, humidity, level etc.) are commonly controlled using PI controllers. In practice, the derivative part is usually switched off due to measurement noise. For pressure and level control in gas tanks, using P controller is sufficient (Bakošová & Fikar, 2008). However, adding derivative part

PD or PID) followed by tuning coefficients of the selected controller type.

Relay

PID controller

Step generator

state error condition (e()=0), type, and parameters of the controlled plant.

According to the Final Value Theorem, the steady-state error

loop gain and wq is a positive constant (Harsányi et al., 1998).

**2.1.2 Principles of controller structure selection based on the plant type**

improves closed-loop stability and steepens the step response rise (Balátě, 2004).

with control action u(t) generated by a PID controller (switch SW in position "1").

**2. PID controller design for performance** 

**2.1 Selection of PID controller structure** 


degree q of the reference signal w(t)=wqtq, i.e.

#### **2.1.3 PID controller structure selection based on plant parametres**

Consider the FOPDT (j=1) and FOLIPDT (j=3) plant models given as GFOPDT=K1e-D1s/[T1s+1] and GFOLIPDT=K3e-D3s/{s[T3s+1]} with following parameters

$$\tau\_1 = \frac{D\_1}{T\_1}; \quad \rho\_1 = K\_1 K\_c; \quad \tau\_3 = \frac{D\_3}{T\_3}; \quad \rho\_3 = \frac{\lim\_{s \to 0} G(s)}{\alpha\_c \left| G(j\alpha\_c) \right|} = \frac{T\_c K\_3 K\_c}{2\pi}; \quad \tau\_3 = \frac{\frac{2}{\pi} + \operatorname{arccg}\sqrt{\rho\_3^2 - 1}}{\sqrt{\rho\_3^2 - 1}} \tag{3}$$

where Kc and c are critical gain and frequency of the plant, respectively. Normed time delay j and parameter j can be used to select appropriate PID control strategy. According to Tab. 1 (Xue et al., 2007), the derivative part is not used in presence of intense noise and a PID controller is not appropriate for plants with large time delays.


Table 1. Controller structure selection with respect to plant model parameters: A: forward compensation suggested, B: forward compensation necessary, C: dead-time compensation suggested, D: dead-time compensation necessary, E: set-point weighing necessary, F: pole-placement

#### **2.2 PID controller design objectives**

Consider the following most frequently used PID controller types: ideal PID (4a), real interaction PID with derivative filtering (4b) and ideal PID in series with a first order filter (4c)

$$\mathcal{G}\_R(\mathbf{s}) = \mathcal{K}\left(1 + \frac{1}{T\_{i^S}} + T\_{d^S}\mathbf{s}\right);\ \mathcal{G}\_R(\mathbf{s}) = \mathcal{K}\left(1 + \frac{1}{T\_{i^S}} + \frac{T\_{d^S}}{1 + \frac{T\_{d^S}}{N}\mathbf{s}}\right);\ \mathcal{G}\_R(\mathbf{s}) = \mathcal{K}\left(1 + \frac{1}{T\_{i^S}} + T\_{d^S}\mathbf{s}\right)\left(\frac{1}{T\_{f^S}\mathbf{s} + 1}\right) \tag{4}$$

In practical cases N8;16 (Visoli, 2006). The PID controller design objectives are:

1. tracking of setpoint or reference variable w(t) by y(t),

2. rejection of disturbance d(t) and noise n(t) influence on the controlled variable y(t).

The first objective called also "servo-tuning" is frequent in motion systems (e.g. tracking required speed); techniques to guarantee the second objective are called "regulator-tuning".

#### **2.3 Performance measures in the time domain**

Performance measures indicating satisfactory quality of setpoint tracking (Fig. 2a) and disturbance rejection (Fig. 2b) are small maximum overshoot and small decay ratio, respectively, given as

$$\eta\_{\text{max}} = 100 \frac{\left| y\_{\text{max}} - y(\infty) \right|}{y(\infty)} \text{ [\%]} ; \quad \delta\_{\text{DR}} = \frac{A\_{i+1}}{A\_i} \tag{5}$$

PID Controller Design for Specified Performance 7

4 – 10) use various weighing of critical parameters thus allowing to vary closed-loop performance requirements. Methods (No. 1 – 10) are applicable for various plant types,

To quickly determine critical parameters Kc and Tc, industrial autotuners apply a relay test (Rotach, 1984) either with ideal relay (IR) or a relay with hysteresis (HR). In the loop in Fig. 1 when adjusting the setpoint w(t) in manual mode and switching SW into "3", a stable limit cycle around y() arises. Due to switching between the levels –M, +M, G(s) is excited by a

**2.4.2 Specification of critical parameters of the plant using relay experiment** 

periodic rectangular signal u(t), (Fig. 3a). Then, c and Kc can be calculated from

4

Fig. 3. A detailed view of u(t) and y(t) to determine critical parameters Kc and Tc

controller coefficients can be found e.g. from the plant step responses (Fig. 4 and 5).

**2.5.1 Specification of FOPDT, IPDT and FOLIPDT plant model parameters** 

Fig. 4. Typical step responses of a) FOPDT; b) IPDT and c) FOLIPDT models

t

From the read-off parameters, transfer functions of individual models have been obtained

D2 1

K2

y y

t

**2.5 Model-based PID controller design with guaranteed performance** 

*c*

where the period and amplitude of oscillations Tc and Ac, respectively, can be obtained from a record of y(t) (Fig. 3b); DB is the width of the hysteresis. Relay amplitude M is usually adjusted at 3%10% of the control action limit. A relay with hysteresis is used if y(t) is corrupted by measurement noise n(t) (Yu, 2006); the critical gain is calculated using (6c).

Steday-state and dynamic properties of real processes are described by simple FOPDT, IPDT, FOLIPDT or SOPDT models. Model parameters further used to calculate PID

According to Fig. 1, the plant step response is obtained by switching SW into "2" and performing a step change in u(t). Plant model parameters are obtained by evaluating the

*c HR*

y

*<sup>M</sup> <sup>K</sup>*

4( 0,5 ) *DB*

*A* 

*c*

t t

(6)

D3 T3 1

K3

t

Ac

*c IR*

*<sup>M</sup> <sup>K</sup> <sup>A</sup>* ; \_

2 *<sup>c</sup> Tc* 

; \_

u M

particular step response (Fig. 4).

T1

D1

y K1

easy-to-use and time efficient.

where y() denotes steady state of y(t). The ratio of two successive amplitudes Ai+1/Ai is measure of y(t) decaying, where i=1...N, and N is half of the number of y() crossings by y(t) (Fig. 2b). A time-domain performance measure is the settling time ts, i.e. the time after which the output y(t) remains within % of its nal value (Fig. 2a); typically =[1%÷5%]y(), DR(1:4;1:2), max(0%;50%). Fig. 2c depicts underdamped (curve 1), overdamped (curve 2) and critically damped (curve 3) closed-loop step responses.

Fig. 2. Performance measures: DR, ts, max and e(); a) setpoint step response; b) load disturbance step response; c) over-, critically- and underdamped closed-loop step-responses

#### **2.4 Model-free PID controller design techniques with guaranteed performance**

Model-free tuning PID controller techniques are used if plant dynamics is not complicated (without oscillations, vibrations, large overshoots) or if plant modelling is time demanding, uneconomical or even unfeasible. To find PID controller coefficients, instead of a full model usually 2-4 characteristic plant parameters are used obtained from the relay test.

#### **2.4.1 Tuning rules based on critical parameters of the plant**

Consider the closed-loop in Fig. 1 with proportional controller. If the controller gain K is successively increased until the process variable oscillates with constant amplitudes, critical parameters can be specified: the period of oscillations Tc and the corresponding gain Kc. If the controller (4a) is considered, coefficients of P, PI and PID controllers are calculated according to Tab. 2, where c=2/Tc is critical frequency of the plant.


Table 2. Controller tuning based on critical parametres of the plant

Rules No. 1 – 3 represent the famous Ziegler-Nichols frequency-domain method with fast rejection of the disturbance d(t) for DR=1:4 (Ziegler & Nichols, 1942). Related methods (No.

; *<sup>i</sup>* <sup>1</sup>

where y() denotes steady state of y(t). The ratio of two successive amplitudes Ai+1/Ai is measure of y(t) decaying, where i=1...N, and N is half of the number of y() crossings by y(t) (Fig. 2b). A time-domain performance measure is the settling time ts, i.e. the time after

=[1%÷5%]y(), DR(1:4;1:2), max(0%;50%). Fig. 2c depicts underdamped (curve 1),

*DR*

*i A A*

t

of its nal value (Fig. 2a); typically

1

y

2 3

(5)

Performance or response

t

max

overdamped (curve 2) and critically damped (curve 3) closed-loop step responses.

A1

Fig. 2. Performance measures: DR, ts, max and e(); a) setpoint step response; b) load disturbance step response; c) over-, critically- and underdamped closed-loop step-responses

A2

**2.4 Model-free PID controller design techniques with guaranteed performance** 

usually 2-4 characteristic plant parameters are used obtained from the relay test.

**2.4.1 Tuning rules based on critical parameters of the plant** 

No. Design method, year Cont-

according to Tab. 2, where c=2/Tc is critical frequency of the plant.

Table 2. Controller tuning based on critical parametres of the plant

Model-free tuning PID controller techniques are used if plant dynamics is not complicated (without oscillations, vibrations, large overshoots) or if plant modelling is time demanding, uneconomical or even unfeasible. To find PID controller coefficients, instead of a full model

Consider the closed-loop in Fig. 1 with proportional controller. If the controller gain K is successively increased until the process variable oscillates with constant amplitudes, critical parameters can be specified: the period of oscillations Tc and the corresponding gain Kc. If the controller (4a) is considered, coefficients of P, PI and PID controllers are calculated

roller K Ti Td

1. (Ziegler & Nichols, 1942) P 0,5Kc - - Quarter decay ratio 2. (Ziegler & Nichols, 1942) PI 0,45Kc 0,8Tc - Quarter decay ratio 3. (Ziegler & Nichols, 1942) PID 0,6Kc 0,5Tc 0,125Tc Quarter decay ratio 4. (Pettit & Carr, 1987) PID Kc 0,5Tc 0,125Tc Underdamped 5. (Pettit & Carr, 1987) PID 0,67Kc Tc 0,167Tc Critically damped 6. (Pettit & Carr, 1987) PID 0,5Kc 1,5Tc 0,167Tc Overdamped 7. (Chau, 2002) PID 0,33Kc 0,5Tc 0,333Tc Small overshoot 8. (Chau, 2002) PID 0,2Kc 0,55Tc 0,333Tc Without overshoot 9. (Bucz, 2011) PID 0,54Kc 0,79Tc 0,199Tc Overshoot max20% 10. (Bucz, 2011) PID 0,28Kc 1,44Tc 0,359Tc Settling time ts13/<sup>c</sup>

Rules No. 1 – 3 represent the famous Ziegler-Nichols frequency-domain method with fast rejection of the disturbance d(t) for DR=1:4 (Ziegler & Nichols, 1942). Related methods (No.

( ) <sup>100</sup> [%] ( ) *y y y*

max

which the output y(t) remains within %

y

t

+

ymax

y(

)

y

ts

4 – 10) use various weighing of critical parameters thus allowing to vary closed-loop performance requirements. Methods (No. 1 – 10) are applicable for various plant types, easy-to-use and time efficient.

#### **2.4.2 Specification of critical parameters of the plant using relay experiment**

To quickly determine critical parameters Kc and Tc, industrial autotuners apply a relay test (Rotach, 1984) either with ideal relay (IR) or a relay with hysteresis (HR). In the loop in Fig. 1 when adjusting the setpoint w(t) in manual mode and switching SW into "3", a stable limit cycle around y() arises. Due to switching between the levels –M, +M, G(s) is excited by a periodic rectangular signal u(t), (Fig. 3a). Then, c and Kc can be calculated from

$$\rho o\_{\mathcal{E}} = \frac{2\pi}{T\_{\mathcal{E}}}; \; K\_{\mathcal{E}\_{-}IR} = \frac{4M}{\pi A\_{\mathcal{E}}}; \; K\_{\mathcal{E}\_{-}HR} = \frac{4(M - 0, 5A\_{\text{DB}})}{\pi A\_{\mathcal{E}}} \tag{6}$$

where the period and amplitude of oscillations Tc and Ac, respectively, can be obtained from a record of y(t) (Fig. 3b); DB is the width of the hysteresis. Relay amplitude M is usually adjusted at 3%10% of the control action limit. A relay with hysteresis is used if y(t) is corrupted by measurement noise n(t) (Yu, 2006); the critical gain is calculated using (6c).

Fig. 3. A detailed view of u(t) and y(t) to determine critical parameters Kc and Tc

#### **2.5 Model-based PID controller design with guaranteed performance**

Steday-state and dynamic properties of real processes are described by simple FOPDT, IPDT, FOLIPDT or SOPDT models. Model parameters further used to calculate PID controller coefficients can be found e.g. from the plant step responses (Fig. 4 and 5).

### **2.5.1 Specification of FOPDT, IPDT and FOLIPDT plant model parameters**

According to Fig. 1, the plant step response is obtained by switching SW into "2" and performing a step change in u(t). Plant model parameters are obtained by evaluating the particular step response (Fig. 4).

Fig. 4. Typical step responses of a) FOPDT; b) IPDT and c) FOLIPDT models

From the read-off parameters, transfer functions of individual models have been obtained

PID Controller Design for Specified Performance 9

According to Haalman (rules No. 24 and 28), controller transfer function GR(s)=L(s)/G(s), where L(s)=0,66e-Ds/(Ds) is the ideal loop transfer function guaranteeing maximum closedloop sensitivity Ms=1,9 to disturbance d(t), (see subsection 2.8.1). For various G(s), various controller structures are obtained. The gain K in rule No. 27 depends on the normed time delay 3=D3/T3 of the FOLIPDT model; for corresponding couples hold: (3;x3)={(0,02;5), (0,053;4); (0,11;3); (0,25;2,2); (0,43;1,7); (1;1,3); (4;1,1)}. Due to integrator contained in IPDT and FOLIPDT models, I-term in the controller structure is needed just to achieve zero

Flexible systems in wood processing industry, automotive industry, robotis, shocks and

For SOPDT model (8b), the relative damping 6(0;1) indicates oscillatory step response.

y

a1

If 6>1, SOPDT model (8a) is used; its parameters are found from the non-oscillatory step

t0=D6

where S=K4(T4+T5+D4) denotes the area above the step response of y(t), and y() is its steady-state value. Parameters of the SOPDT model (8b) can be found from evaluation of 2-4

Quality of identification improves with increasing number of read-off amplitudes N. If N>2 several values 6, T6 and D6 are obtained and their average is taken for further calculations. Tab. 5 summarizes useful tuning formulae for both oscillatory and non-oscillatory systems

*<sup>D</sup>* ; 0,33 0,7

<sup>1</sup> 1,259 *t t*

; 6 1 <sup>1</sup>

1

*i*

*N*

1 1

*<sup>N</sup> D t tt N*

2

*i N*

(10)

*<sup>C</sup>* ; 2 4 ( )

*<sup>S</sup> C D*

*<sup>y</sup>* (9)

a3

A3

<sup>4</sup> 0,516 1,067 *t t*

periods of step response oscillations (Fig. 5b) using following rules (Vítečková, 1998)

; 2 6 6 1 1

1 *T tt <sup>N</sup> <sup>N</sup>* 

 ; <sup>6</sup>

*K e G s*

A1

*SOPDT*

6 2 2 6 66 ( ) 2 1

*T s Ts* 

a2 A2

t1 t2 t3

*D s*

(8)

t

 

vibrations damping are often modelled by SOSPTD models with transfer functions

4 4 4 5 ( ) 1 1

y(

)

*D s*

*Ts Ts* 

Fig. 5. Step response of SOPDT model: a) non-oscillatory, b) oscillatory

t

steady-state error e() under steady-state disturbance d().

**2.5.4 Tuning formulae for SOPDT plant models**

response in Fig. 5a using the following relations

*T CCC* ; 0,33 0,7

 2 2 4,5 2 2 1 <sup>1</sup> <sup>4</sup>

1

2 2 1

*i i i i*

*a a a a*

ln

ln

with SOPDT model properties.

2

**S** 

t0,33 t0,7

y

D4

0,33y()

0,7y()

y()

6

*K e G s*

*SOPDT*

$$\mathbf{G}\_{\rm FOPDT}(\mathbf{s}) = \frac{\mathbf{K}\_1 \mathbf{e}^{-D\_1 s}}{T\_1 \mathbf{s} + \mathbf{1}}; \ \mathbf{G}\_{\rm IPDT}(\mathbf{s}) = \frac{\mathbf{K}\_2 \mathbf{e}^{-D\_2 s}}{s}; \ \mathbf{G}\_{\rm FOLIPDT}(\mathbf{s}) = \frac{\mathbf{K}\_3 \mathbf{e}^{-D\_3 s}}{s \left(T\_3 s + 1\right)}\tag{7}$$
