Open access peer-reviewed chapter

Advanced Methods of PID Controller Tuning for Specified Performance

By Štefan Bucz and Alena Kozáková

Submitted: June 8th 2017Reviewed: February 28th 2018Published: September 12th 2018

DOI: 10.5772/intechopen.76069

Downloaded: 807

Abstract

This chapter provides a concise survey, classification and historical perspective of practice-oriented methods for designing proportional-integral-derivative (PID) controllers and autotuners showing the persistent demand for PID tuning algorithms that integrate performance requirements into the tuning algorithm. The proposed frequency-domain PID controller design method guarantees closed-loop performance in terms of commonly used time-domain specifications. One of its major benefits is universal applicability for both slow and fast-controlled plants with unknown mathematical model. Special charts called B-parabolas were developed as a practical design tool that enables consistent and systematic shaping of the closed-loop step response with regard to specified performance and dynamics of the uncertain controlled plant.

Keywords

  • PID controller tuning
  • robust performance indices
  • B-parabolas
  • closed-loop
  • performance assessment
  • robust performance

1. Introduction

How to tune a controller for any control application quickly and appropriately? This question raised in 1942 is still up to date and constantly occupies the automation community worldwide. The answer is very intricate; its intricacy is comparable with the open hitherto unresolved Hilbert problems known from mathematics.

Will the PID controllers, historically the oldest but currently still the most used ones, control industrial processes in the near and far future? Based on the increase of the number of PID tuning methods from 258 to 408 during 2000–2005, a positive response can be assumed [23]. The remarkably simple ability of the PID controller to generate a difference equation using the present, past and future values of the control error is often projected into the philosophical understanding [1] and forecast this controller a long-term perspective.

Beginnings of PID controllers date back to 1935 when the Taylor Instruments Companies launched their pneumatic controller with a derivative channel [1]. Owing to rapid developments in the control theory, it was supposed that the conventional PID controllers would be gradually replaced by advanced ones; however, this did not come to pass mainly due to the simple PID structure and its commercial usability in practice. For 83 years, control loop designers preferred the PID controllers for their outstanding ability to eliminate the control error using the integrator, their ability to improve the performance using the “trend” of the controlled variable through the derivative channel and for many other benefits. PID controllers are important parts of distributed control systems, predictive control structures; their coefficients are often adapted by means of fuzzy and neural control and set by genetic algorithms [20, 21, 35]. In multiloop control structures, they are able to stabilize unstable objects and difficult-to-control systems. The 46 existing PID variants and reported 408 diverse tuning methods are a good prerequisite for achieving a satisfactory performance in simple as well as demanding industrial applications [23, 25].

PID controllers are widely applied in technological processes of heavy and light industries, for example in control of tension in the roll during paper winding, boiler temperature, chemical reactor pressure, lathe spindle position in metalworking, and so on; they can be found in modern cars controlling combustion control or vehicle dynamics [9], valve opening and robotic arm position. In the interconnected power system, they are used to control turbine power and speed in both primary and secondary regulation of active power and network frequency. Being easy to implement on both Arduino and Raspberry Pi platforms allows them to be used in mobile “unplugged” applications as well.

Commercial applicability of PID controllers is confirmed by studies referring that more than 90% out of all installed controllers in industrial control loops are PID controllers [36]. The alarming fact, however, is that only 20% of them are tuned correctly, and in 30% of all PID applications, the regulation is unsuitable due to an incorrect selection of synthesis method. Another 30% of poor performance is due to ignorance of nonlinear properties of actuators, and the remaining 20% represent an inadequate choice of sampling period or poor signal filtering [5]. Some controllers not only do not provide the required performance yet often even stability of the control loop being operated only in open loop and manually switched off by the service staff when approaching the setpoint [9]. According to other statistics, 30% of controllers operate in manual mode and require continuous fine-tuning and supervising by the process technologist. A 25% of PID applications use coefficients pre-set by the manufacturer with no update of their values with respect to the particular process [47].

Therefore, a natural requirement for innovative PID tuning methods has come up to ensure the specified performance [19, 22] in terms of the maximum overshoot and settling time not only for processes with constant parameters but for their perturbed types. In this chapter, a novel original robust PID tuning method is presented; hopefully, it will help reverse the above mentioned unfavorable statistics of incorrectly tuned PID controllers.

2. PID controller design for industrial processes for performance

Despite the fact that there are more than 11,000 PID controllers in 46 variants operating in industrial processes [23], mostly three basic forms are used to control industrial processes: the ideal (textbook) PID controller, the real PID controller with derivative filter, and the ideal PID controller in series with the first-order filter given by the following transfer functions, respectively.

GRs=K1+1Tis+Tds,GRs=K1+1Tis+Tds1+TdNs,GRs=K1+1Tis+Tds1Tfs+1,E1

where K is the proportional gain, Ti and Td are the integral and the derivative time constants, respectively, Tf is the filter time constant and N∈<8,16> in practical applications [37, 38]. 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).

Time response of the controlled variable y(t) is modifiable by parameters K, 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 performance criteria.

Consider the control loop in Figure 1 with control action u(t) generated by the PID controller (switch SW in position “1”).

Figure 1.

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

A controller design is a two-step procedure consisting of controller structure selection (P, PI, PD or PID) followed by tuning coefficients of the selected controller type.

2.1. PID controller structure selection

An appropriate structure of the controller GR(s) is usually selected with respect to:

  • zero steady-state error condition (e(∞) = 0),

  • type of the controlled plant,

  • parameters of the controlled plant.

2.1.1. PID controller structure selection based on zero steady-state error condition

Consider the unity feedback control loop (Figure 1), where G(s) is the controlled system. According to the final value theorem, the steady-state error

e=lims0sEs=lims0s11+LsWs=q!wqlims0sνqsν+KLE2

is zero if the integrator orders νL = νS + νR in the open-loop L(s) = G(s)GR(s) is greater than the order q of the reference signal w(t) = wqtq, i.e.,

νL>q,E3

where νS and νR are integrator orders of the plant and the controller, respectively, KL is the open-loop gain and wq is a positive constant [46].

2.1.2. PID controller structure selection based on the plant type

Industrial process variables (e.g., position, speed, current, temperature, pressure, humidity, level) 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 a P-controller is sufficient [3]. However, adding the derivative part improves closed-loop stability and steepens the step response rise time.

2.1.3. PID controller structure selection based on plant parameters

Consider the FOPDT (j = 1) and FOLIPDT (j = 3) plant models, respectively given as GFOPDT=K1eD1s/T1s+1and GFOLIPDT=K3eD3s/sT3s+1with the parameters given as follows:

μ1=D1T1;ϑ1=K1Kc;μ3=D3T3;ϑ3=lims0sGsωcGjωc=TcK3Kc2π;μ3=2π+arctgϑ321ϑ321,E4

where Kc and ωc are critical gain and critical frequency of the plant, respectively, the normalized time delay μj and the parameter ϑj can be used to select appropriate PID control strategy. According to Table 1 [46], the derivative part is not used in presence of intense noise and a PID controller is not appropriate for plants with large time delays.

Ranges for μ and ϑControl strategy
No precise control necessaryPrecise control needed
High noiseLow saturationLow measurement noise
μ1 > 1; ϑ1 < 1.5II + B + CPI +B + CPI +B + C
0.6 < μ1 < 1; 1.5 < ϑ1 < 2.25I or PII + API + A(PI or PID) + A + C
0.15 < μ1 < 0.6; 2.25 < ϑ1 < 15PIPIPI or PIDPID
μ1 < 0.15; ϑ1 > 15 or μ3 > 0.3; ϑ3 < 2P or PIPIPI or PIDPI or PID
μ3 < 0.3; ϑ3 > 2PD + EFPD + EPD + E

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 weighting necessary, F: pole-placement.

2.2. Performance measures in PID controller design

Performance measures for industrial control loops can be expressed both in the time and the frequency domains. The time-domain performance indicators allow to directly expressing the desired process parameter, whereas the frequency-domain performance indicators can be used as PID tuning parameters.

2.2.1. Performance measures in the time-domain

In the time-domain, satisfactory setpoint tracking (Figure 2a) and disturbance rejection (Figure 2b) are indicated by small values of maximum overshoot and a decay ratio, respectively, given as

ηmax=100ymaxyy%;δDR=Ai+1Ai,E5

where y(∞) denotes the steady-state value of y(t) [4]. A measure of the y(t) response decay is the ratio of two successive amplitudes Ai+1/Ai, where i = 1…N, and N is half the number of points of intersections of y(∞) and y(t). The settling time ts is the time after which the output y(t) remains within ±ε% of its final value (Figure 2b); typically ε = [1%÷5%]y(∞), δDR∈(1:4;1:2), ηmax∈(0%;50%). Figure 2c depicts underdamped (plot 1), overdamped (plot 2) and critically damped (plot 3) closed-loop step responses.

Figure 2.

Performance measures: δDR, ts, ηmax and e(∞); (a) setpoint step response; (b) load disturbance step response; (c) over-, critically- and underdamped closed-loop step-responses.

2.2.2. Performance measures in the frequency-domain

The most frequent parameters for PID tuning are the following performance measures [1]:

  • ϕM and GM: phase and gain margins, respectively,

  • Ms and Mt: maximum peaks of magnitudes of the sensitivity function S(jω) and complementary sensitivity function T(jω), respectively,

  • λ: required closed-loop time constant.

If a designed controller GR(jω) guarantees, that |S(jω)| or |T(jω)| do not exceed prespecified values Ms or Mt, respectively, defined by

Ms=supωS=supω11+L;Mt=supωT=supωL1+LE6

for ω∈⟨0,∞), then the Nyquist plot of the open-loop transfer function L(s) = G(s)GR(s) avoids the respective circles MS or MT each given by its center and radius as follows:

CS=1j0,RS=1Ms;CT=Mt2Mt21j0,RT=Mt1Mt2.E7

By avoiding the Nyquist plot of L(s) to enter the circles corresponding to MS or MT, a safe distance from the critical point is kept (Figure 3a). Typical |S(jω)| and |T(jω)| plots for properly designed controller are in Figure 3b. The disturbance d(t) is sufficiently rejected if Ms∈(1.2;2). The reference w(t) is properly tracked by the process output y(t) if Mt∈(1.3;2.5). With further increasing of Mt the closed-loop tends to be oscillatory.

Figure 3.

(a) Definition and geometrical interpretation of ϕM and GM in the complex plane; (b) sensitivity and complementary sensitivity magnitudes |S(jω)|, |T(jω)| and performance measures Ms, Mt.

From Figure 3a results, that by increasing open-loop phase margin ϕM the gain crossover L(jωa*) on the unit circle M1 moves away from the critical point (−1,j0); similarly by increasing open-loop gain margin GM the phase crossover L(jωf*) moves away from (−1,j0). Therefore, the stability margins ϕM or GM given by

φM=180°+argLωa;GM=1LjωfE8

are frequently used performance measures, their typical values are ϕM∈(20°;90°), GM∈(2;5). Relations between individual stability margins and respective magnitude peaks are given by the following inequalities

φM2arcsin12Ms;φM2arcsin12Mt;GMMsMs1;GM1+1Mt.E9

The point in which the Nyquist plot L(jω) touches the MT circle defines the closed-loop resonance frequency ωMt.

2.3. PID controller design methodologies for performance

When synthesizing a control loop, if the controller type is already known and the designer has just to select a suitable method to appropriately adjust its coefficients, we speak about PID controller tuning methods. Controller design is a more complex problem which includes determining controller structure and then calculating its parameters. When setting coefficients of industrial PID controllers {K, Ti, Td}, basically the following procedures are applied:

  1. Trial-and-error methods are based on closed-loop experiments [1]. Controller parameters settings are based on observation of the response to reference or disturbance changes with the assistance of an expert, or the design is driven by empirical rules. The control-loop synthesis is time-consuming and its success is not guaranteed.

  2. Analytical methods are used to generate a control law based on the mathematical model of the plant; the plant model is obtained from first principles or via experimental identification. The success of these methods depends on the accuracy of the mathematical model of the plant, and is not always achievable in practical cases (e.g., for a cement kiln).

  3. Classical tuning methods use only a limited number of characteristic parameters of the plant obtained from the step response or critical system parameters [11, 27, 31]. Their main advantage is a simple and short calculation of controller parameters. The control objective is to provide a satisfactory response to reference change, or disturbance rejection and often their combination. The main drawback is that the designer cannot influence the performance by means of the adjustable parameters of the algorithm. Also, the resulting closed-loop response may not be satisfactory if the step response of the plant is nonmonotonic, or when the plant has nonminimum-phase dynamics or large time delay. Most of these methods are implemented in autotuners of industrial PID controllers [1].

  4. Autotuners are a modern and convenient means for adjusting coefficients of industrial controllers [33, 34, 49]. They implement a two-step algorithm of automatic acquisition of characteristic parameters of the controlled process followed by automatic calculation and adjustment of the controller coefficients. After activating the autotuning function on the industrial controller, the control-loop synthesis is performed automatically in a very short time. The ABB, Emerson, Fischer-Rosemont, Foxboro, Honeywell, Siemens, Yokogava, or ZPA controllers have a built-in PID autotuning function implemented on a microcomputer [47]. In many situations, however, these methods are unreliable because of the imperfection of the plant identification algorithm and the subsequent controller design.

  5. Robust PID controller tuning methods for specified performance represent a modern area of industrial control-loop synthesis. They improve the PID tuning methods by providing stability and required performance also for processes with variable parameters. The controller tuned only by conventional method is just “intuitively” invariant against perturbations of the controlled plant; robust operation of the control loop is usually possible only for small changes of plant parameters. The major disadvantage of these methods is that the control law is not based on the knowledge of uncertainties of the controlled object and a further research on their possible expansion is needed. The proposed original method which eliminates this drawback, its theoretical analysis and verification on benchmark examples are the core of the chapter.

2.4. PID controller tuning methods for performance

Tuning methods are commonly used engineering tool for the synthesis of industrial control loops as they do not require a full knowledge of the mathematical description of the controlled plant. This differentiates them from analytical methods which, on the contrary are based on a precise knowledge of the mathematical model of the controlled system. In the tuning methods, the controlled process is considered as a black-boxwhich is to be revealed only to such extent that the controller synthesis is successful and the control objectivesare achieved. Thus, only those characteristic parametersof the unknown plant have to be acquired via appropriate identificationthat are inevitable for the PID controller design. In this way, the PID controllercoefficients can be obtained in a relatively short time. The implicit knowledge about the controlled systemand the ambient influences affect the choice of the PID coefficients calculation method. According to the way of using the identified data of the controlled plant, the tuning methods are classified into as follows:

  1. model-free PID controller tuning methods,

  2. model-based PID controller tuning methods.

Percentage proportions of commonly used PID controller tuning methods are presented in Figure 4.

Figure 4.

Percentages of commonly used PID controller tuning methods.

Approximately 12% out of all tuning methods are model-free methods, 88% (app. seven times more) are model-based ones. A 37% portion belongs to PID controllers of FOLPDT (first order lag plus dead time) system models which are the most commonly used approximation of plant dynamics in industries (thermal plants, chemical and woodworking industries). Equal 9% shares belong to IPDT (integral plus dead time) and FOLIPDT (first order lag integral plus dead time) models with integral behavior encountered mainly in drives and power industry in modeling mechanical subsystems of rotating machines, valves and servo-systems and 18% of algorithms are used to control plants with SOSPDT (second order system plus dead time) models [26]. Controllers for other system types are tuned by methods from the 15% portion.

2.4.1. Model-free PID controller tuning with guaranteed performance

There are such PID tuning algorithms, which can be applied without any knowledge about the unknown plant model. These methods yielding PID coefficients for systems with a general, unknown model are known as “Non-Model Specific”, “Model-Free” or “Rule-Based” Methods. Their basic feature is that the identified characteristic parameters of the unknown system appear directly in the PID coefficient tuning rules. They are very popular among practitioners due to a high flexibility and ability to control a wide class of systems. The respective algorithms have been tested on benchmark examples, the control objectives can be expressed by empirical rules. They are simply algorithmizable for application in industrial autotuners. A seven-step flow diagram of a direct tuning method is depicted in Figure 5.

Figure 5.

Flow chart of the direct engineering PID tuning method.

The oldest direct-type engineering method is the well-known Ziegler-Nichols frequency method (1942) [48]. The control objectiveis a rapid disturbance rejection so that each amplitude of the oscillatory response to disturbance step change is only a quarter of the previous amplitude. The method is based on two identifiedcharacteristic parametersof the unknown plant: the critical frequency ωc = 2π/Tc and the critical gain Kc used for calculation of the coefficients of P, PI and PID controllers. The first characteristic parameter provides basic information on plant dynamics, while the value of the second parameter indicates the degree-of-stability of the plant. PID controller parameters according to the Ziegler-Nichols method are calculated using the algorithmPZN = 0.6Kc, TiZN = 0.5Tc = π/ωc, TdZN = 0.125Tc = 0.25π/ωc, in which the characteristic parameters {Kc, ωc} are directly included.

2.4.1.1. Trial-and-error tuning methods

When first PID controllers were developed in the period 1935–1942, no tuning methods were available at the market. The controller “design” consisted in experimenting with control loops without considering any relationship between plant parameters and controller coefficients. Acquired experience, however, was generalized giving rise to empirical trial-and-error tuning method that consist of three main steps:

  1. Turning off the integral and derivative parts of the PID controller and increasing the gain until the closed-loop oscillates with constant amplitudes, then adjusting the gain at half of this value.

  2. Decreasing the integral time until oscillations with constant amplitude are obtained, then adjusting the integral time at a treble of this value.

  3. Increasing the derivative time until oscillations with constant amplitude are obtained, then adjusting the derivative time at a third of this value.

The set of these rules of thumb is still being used in practice to roughly tune industrial PID controllers and is considered as a predecessor of all engineering tuning methods. In 1942, two direct tuning methods were published and authored by Ziegler and Nichols [48], employees of the Taylor Instrument Companies producing PID controllers. The first one is time-domain method; according to it, the PID coefficients are calculated using the effective time delay and the effective time constant of the step response of the industrial plant. The frequency-domain method uses the critical gain Kc and the period of critical oscillations Tc to calculate the PID coefficients according to the relations ΘPID = (P, Ti, Td) = (α1Kc, α2Tc, α3Tc), where the weights of critical parameters are (α1, α2, α3) = (0.6, 0.5, 0.125).

2.4.1.2. Tuning rules based on ultimate parameters of the industrial process

Due to its simplicity, the Ziegler-Nichols frequency-domain methods are still used in industrial autotuners in the original version, although they have undergone various modifications during the last 70 years of its existence. Due to the technological development after the industrial revolution and major electrification, PID tuning for stability was no more sufficient because a fast setpoint attainment could bring about important savings of time and money and accelerate the entire production process. More and more demanding requirements on control performance were formulated, and an intense demand for effective tuning methods guaranteeing required performance has arisen.

As a rule, application of Ziegler-Nichols methods usually leads to oscillatory closed-loop responses; hence, many scientists have become interested in their possible improvement. Forty-two modifications of the Ziegler-Nichols frequency method were developed in the period from 1967 to 2010. They differ from the classical algorithm in using various other combinations of the weights (α1, α2, α3). An overview of selected model-free methods is given in Table 2.

No.Design method, yearControllerKTiTdPerformance
1.Ziegler and Nichols, 1942 [48]P0.5KcQuarter decay ratio
2.Ziegler and Nichols, 1942 [48]PI0.45Kc0.8TcQuarter decay ratio
3.Ziegler and Nichols, 1942 [48]PID0.6Kc0.5Tc0.125TcQuarter decay ratio
4.Pettit and Carr, 1987 [27]PIDKc0.5Tc0.125TcUnderdamped
5.Pettit and Carr, 1987 [27]PID0.67KcTc0.167TcCritically damped
6.Pettit and Carr, 1987 [27]PID0.5Kc1.5Tc0.167TcOverdamped
7.Chau, 2002 [16]PID0.33Kc0.5Tc0.333TcSmall overshoot
8.Chau, 2002 [16]PID0.2Kc0.55Tc0.333TcNo overshoot
9.Bucz, 2011 [6]PID0.54Kc0.79Tc0.199TcOvershoot ηmax≤20%
10.Bucz, 2011 [6]PID0.28Kc1.44Tc0.359TcSettling time ts≤13/ωc

Table 2.

Model-free PID controller tuning rules based on critical plant parameters.

Tuning rules No. 1–3 are the well-known Ziegler-Nichols frequency-domain method which objective is a fast rejection of the disturbance d(t) and δDR = 1:4. In the complex plane interpretation (Figure 6), the method corresponds to shifting the critical point C = [−1/Kc + j0], into the points CP = [−0.5 + j0], CPI = [−0.45 + j0.0896] and CPID = [−0.6-j0.28] using respectively P, PI and PID controllers tuned according to Table 2. Put simply, the open-loop Nyquist plot is shaped into a sufficient distance from the limit of instability specified by the point (−1.j0).

Figure 6.

Moving the critical point C = [−1/Kc + j0] of the plant using P, PI and PID controllers designed by Ziegler-Nichols frequency-domain method for critical frequency ωc of the plant.

Related methods (No. 4–10) use various weighting of critical parameters thus allowing to vary the closed-loop performance requirements (see the last column in Table 2). All presented methods (No. 1–10) are applicable for various plant types, easy-to-use and time efficient.

2.4.1.3. Specification of critical parameters of the plant using relay experiment

In autotuners of industrial controllers, a relay test [29] using an ideal relay (IR) or a hysteresis relay (HR) is used to quickly determine the plant critical parameters Kc and Tc. In the manual mode, after setting the nominal setpoint w(t) and switching the SW to position “3”, a stable limit cycle around the nominal working point y(t) arises in the control loop in Figure 1. As a result of switching between the −M, +M relay levels, the controlled system G(s) is excited by a rectangular periodic signal u(t) (Figure 7a). The critical frequency ωc and the critical gain Kc are calculated as follows:

ωc=2πTc;Kc_IR=4MπAc;Kc_HR=4M0.5ΔDBπAc,E10

where the period Tc and the amplitude Ac of critical oscillations are read off from y(t) of the recorded limit cycle (Figure 7b); ΔDB is the width of the hysteresis plot, the relay amplitude M is chosen as (3÷10)% of the control u(t) limits. A typical limit cycle is depicted in Figure 7b. A hysteresis relay is used if y(t) corrupted by a noise n(t) [47].

Figure 7.

Determination of critical parameters Kc and Tc of the controlled plant from the limit cycle.

The advantage of these methods is their applicability for different types of systems, simplicity and the short time needed for the controller design of the—approx. (3÷4)Tc.

2.4.2. Model-based PID controller tuning with guaranteed performance

In these methods, the identified characteristics of the unknown system are used to create its typical model, and the controller design algorithm is derived for this particular model. Formulas for calculation of the controller coefficients include process model parameters that are function of the identified process data. Each method works perfectly for the system whose model has been used in the design algorithm. However, if the system is approximated differently, the achieved performance may be impaired or even insufficient. The advantage is that control objectives can be clearly defined and expressed using analytical relationships (e.g., it is possible to derive the relationship for maximum overshoot of the step response). A small flexibility due to the “tailor-made” design for one type of model limits the widespread application of these methods in autotuners of industrial controllers.

PID tuning algorithms of indirect tuning methods have two more steps compared with direct methods, as shown in the flow chart in Figure 8. When choosing the procedure for creating a typical model., it is important how the implicit information about the controlled system is considered (if we deal with a driving system, a thermal process, a mechanical or pneumatic system, etc.). If the typical model for the given controlled system has already been selected, the model parameters are calculated in step.and subsequently used in calculation of the PID coefficients.

Figure 8.

Flow chart of the indirect engineering method for PID tuning.

2.4.2.1. Specification of FOLPDT, IPDT and FOLIPDT plant model parameters

The static and dynamic properties of most technological processes can be expressed by one of the FOLPDT, IPDT, FOLIPDT, or SOSPDT models. Model parameters are identified from the recorded step response of the controlled system (Figure 9) and are further used in calculation of PID controller coefficients. According to Figure 1, step response of the controlled process is obtained by switching SW into position “2” and performing step change in u(t).

Figure 9.

Typical step responses of (a) FOLPDT; (b) IPDT and (c) FOLIPDT models.

Transfer functions of the model are found from the step response parameters according to Figure 9.

GFOLPDTs=K1eD1sT1s+1;GIPDTs=K2eD2ss;GFOLIPDTs=K3eD3ssT3s+1.E11

2.4.2.2. PID controller tuning formulas for FOLPDT models

The FOPDT model (11a) is used to approximate dynamics of chemical processes, thermal plants, production processes, and so on. To calculate the P, PI and PID controller, coefficients based on the parameters of the FOLPDT model of the controlled system, the tuning formulas in Table 3 can be used.

No.Design method, year, control purposeControllerKTiTdPerformance
11.Ziegler and Nichols, 1942 [48]P1/ϑ1Quarter decay ratio
12.Ziegler and Nichols, 1942 [48]PI0.9/ϑ13D1
13.Ziegler and Nichols, 1942 [48]PID1.2/ϑ12D10.5D1
14.Chien et al., 1952, regulator tuning [18]PI0.6/κ14D1ηmax = 0%, D1/T1∈(0.1;1)
15.Chien et al., 1952, regulator tuning [18]PID0.95/ϑ12.38D10.42D1
16.Chien et al., 1952, regulator tuning [18]PI0.77/ϑ12.33D1ηmax = 20%, D1/T1∈(0.1;1)
17.Chien et al., 1952, regulator tuning [18]PID1.2/ϑ12D10.42D1
18.Chien et al., 1952, servo tuning [18]PI0.35/ϑ11.17D1ηmax = 0%, D1/T1∈(0.1;1)
19.Chien et al., 1952, servo tuning [18]PID0.6/ϑ1D10.5D1
20.Chien et al., 1952, servo tuning [18]PI0.6/ϑ1D1ηmax = 20%, D1/T1∈(0.1;1)
21.Chien et al., 1952, servo tuning [18]PID0.95/ϑ11.36D10.47D1
22.ControlSoft Inc., 2005 [23]PID2/K1T1 + D1max(D1/3;T1/6)Slow loop
23.ControlSoft Inc., 2005 [23]PID2/K1T1 + D1min(D1/3;T1/6)Fast loop

Table 3.

PID tuning rules based on FOPDT model, ϑ1 = K1D1/T1 is the normalized process gain.

2.4.2.3. PID controller tuning formulas for IPDT and FOLIPDT models

While dynamics of slow technological processes (polymer production, heat exchange, etc.) can be approximated by an IPDT model (11b), electromechanical subsystem of rotating machines and servo drive objects are typical examples for using a FOLIPTD model [42] (11c) (Table 4).

No.Design method, year, modelControllerKTiTdPerformance
24.Haalman, 1965, IPDT model [12]P0.66/(K2D2)Ms = 1.9
25.Ziegler and Nichols, 1942, IPDT model [48]PI0.9/(K2D2)3.33D2Quarter decay ratio
26.Ford, 1953, IPDT model [10]PID1.48/(K2D2)2D20.37D2Decay ratio 1:2.7
27.Coon, 1956, FOLIPDT model [8]Px3K3T3+D3Quarter decay ratio
28.Haalman, 1965, FOLIPDT model [12]PD0.66/(K3D3)T3Ms = 1.9

Table 4.

PID tuning rules based on IPDT and FOLIPDT model parameters.

The gain K in the rule No. 27 is variable with respect to the normalized time delay υ3 = D3/T3 of the FOLIPDT model; for the corresponding pairs holds: (υ3;x3) = {(0.02;5), (0.053;4); (0.11;3); (0.25;2.2); (0.43;1.7); (1;1.3); (4;1.1)}.

2.4.2.4. PID controller tuning formulas for SOSPDT plant models

Flexible systems in wood processing industry, automotive industry, robotics, shocks and vibrations damping are often modeled by SOSPDT models with transfer functions

GSOSPDTs=K4eD4sT4s+1T5s+1;GSOSPDTs=K6eD6sT62s2+2ξ6T6s+1,E12

where for SOSPDT model (12b) the relative damping ξ6∈(0;1) indicates oscillatory step response.

If ξ4 > 1, SOSPDT model (12a) is used; its parameters are found from the nonoscillatory step response in Figure 10a using the following relations

T4.5=12C2±C224C12;D4=t0.330.516t0.71.067;C1=t0.33t0.71.529;C2=Sy,E13

where S = K4(T4 + T5 + D4) is the area above the step response of the process output y(t), and y(∞) is its steady-state value.

Figure 10.

Step response of (a) nonoscillatory, (b) oscillatory SOSPDT model.

Parameters of the SOSPDT model (12b) can be found from evaluation of 2–4 periods of step response oscillations (Figure 10b) using following rules [39]

ξ6=lnai+1aiπ2+ln2ai+1ai;T6=1ζ62πNtN+1t1;D6=1Ni=1NtiN+12tN+1t1.E14

Quality of identification improves with increasing number N of read-off amplitudes. If N > 2 several values ξ6, T6 and D6 are obtained, and their average is taken for further calculations. Table 5 summarizes useful tuning formulas for both oscillatory and nonoscillatory systems with SOSPDT model properties.

Table 5.

Tuning rules based on SOSPDT model parameters.

Using tuning methods shown in Tables 25, achieved performance is a priori given by the particular method (e.g., quarter decay ratio when using Ziegler-Nichols methods No. 11–13 in Table 3) or guarantees performance however not specified by the designer (e.g., in Chen method No. 33 in Table 5 gain margin GM = 1.96, phase margin ϕM = 44.1° and maximum peak Ms = 1.5 of the sensitivity to disturbance d(t)).

2.4.2.5. PID controller tuning formulas for unstable FOLPDT models

Minimization of performance indices can be applied also for unstable FOLPDT models

GFOLPDT_USs=K1eD1sT1s1E15

leading to simple tuning rules for PID controller (1a) (No. 34–38 in Table 6). Tuning rules No. 37 and 38 for PID controller (1c) show that settling time ts increases with growing normalized time delay ν1 = D1/T1 of the FOLPDT model (15).

No.Method, yearKTiTdTfPerformance
34.Visioli, 2001, Regulator tuning [36]1.37ν1/K12.42T1ν11.180.60T1Minimum ISE
35.Visioli, 2001, Regulator tuning [36]1.37ν1/K14.12T1ν10.900.55T1Minimum ISTE
36.Visioli, 2001, Regulator tuning [36]1.70ν1/K14.52T1ν11.130.50T1Minimum IST2E
37.Chandrashekar et al., 2002 [15]10.3662/K10.3874T10.0435T10.0134T1ts = 0.1 T1: ν1 = 0.1
38.Chandrashekar et al., 2002 [15]2.0217/K14.65T10.2366T10.0696T1ts = 0.8 T1: ν1 = 0.5

Table 6.

Tuning rules for unstable FOPDT model.

2.5. PID controller design for specified performance

The main benefit of these methods consists of that all tuning rules are based on a single tuning parameter that enables to systematically affect the closed-loop performance by step response shaping [32].

2.5.1. PID controller tuning formulas with performance specification

Table 7 shows open formulas for PID controller design; their tuning is carried out with respect to closed-loop performance specification.

No.Design method, year, modelKTiTd
39.Hang and Åström, 1988, Nonmodel [13]KcsinφMTc1cosφMπsinφMTc1cosφM4πsinφM
40.Rotach, 1994, Nonmodel [28]MtGjωMtMt212ωMt2dargGωMtdωMt12dargGωMtdωMt
41.Wojsznis et al., 1999, FOPDT [45]KccosφMGMTcπtgφM+1+tg2φMTc4πtgφM+1+tg2φM
42.Morari and Zafiriou, 1989, FOPDT [22]T1+0,5D1K1λ+D1T1+12D1T1D12T1+D1
43.Chen and Seborg, 2002, FOPDT [17]T12+T1D1λT12λ+L2T12+T1D1λT12T1+L1-

Table 7.

PID design for specified performance based on tuning parameters ϕM, GM, Mt and λ.

Rules No. 39–43 consider tuning of ideal PID controller (1a). To apply the Rotach method [29], knowledge of the plant magnitude |G(jω)| is supposed as well as of the roll-off of the phase plot argG(ω) at ω = ωMt, where the maximum peak Mt of the complementary sensitivity is required. Method No. 42 is based on the so-called λ-tuning, where the resulting closed-loop is expressed as a 1st order system with time constant λ; this rule considers real PID controller (1b) with filtering constant in the derivative part Tf = 0.5λD1/(1 + D1), where λ is to be chosen so as to meet following conditions: λ>0.25D1; λ>0.25T1 [22]. The λ-tuning technique is used also in the rule No. 43 to design interaction PI controller.

2.5.2. Closed-loop performance evaluation under PID controller tuning

Phase margin ϕM is the most widespread performance measure in PID controller design. Maximum overshoot ηmax and settling time ts of the closed-loop step response are related with ϕM according to Reinisch relations

ηmax=0.91φM+64.55forφM38°71°1.53φM+88.46forφM12°38°;ηmax=100e2πb2Mt;tsπωa4πωaE16

valid for second-order closed-loop with relative damping ωa*∈(0.25;0.65) where ωa* is the gain crossover frequency [14]. Relations

ηmax1001.18MtT0T0%;ts3ωa forMt1.31.5E17

are general for any order of the closed-loop T(s); if the controller has the integral part then |T(0)| = |T(ω = 0)| = 1 [14].

The engineering practice is persistently demanding for PID controller design methods that simultaneously guarantee several performance criteria [24], especially the maximum overshoot ηmax and the settling time ts. However, we ask the question: how to suitably transform the abovementioned engineering requirements into frequency-domain specifications applicable for PID controller coefficients tuning? The response can be found in Section 3 in which a novel original PID controller design method is presented.

3. PID controller design for specified performance based on harmonic excitation

The proposed original method [6] enables to guarantee required closed-loop performance for a whole family of plants specified by the uncertainty description. The core of it is the recently developed PID controller design method based on external harmonic excitation [7]—a two-step PID tuning method for performance specified in terms of maximum overshoot ηmax and settling time ts.

In the first step, the plant is identified using external harmonic excitation signal (a sinusoid) with the frequency ωn. In the second step, two developed PID controller design approaches can be applied:

  1. the approach based on guaranteed phase margin ϕM suitable for nonintegrating systems with/without time delay and for integrating systems as well;

  2. the approach based on guaranteed gain margin GM suitable for nonintegrating systems with unstable zero.

For the ϕM–based approach, the specified performance is achieved by means of developed quadratic dependences ηmax = f(ϕMn) and ts = f(ϕMn) parameterized by ωn; the corresponding plots are called B-parabolas. For the GM–based approach similar quadratic dependences for both the maximum overshoot ηmax = f(GMn) and the settling time ts = f(GMn) were constructed. These approaches enable to achieve fulfillment of the following performance measures (ωc is the plant ultimate frequency):

  • for plants without integration behavior: ηmax∈⟨0%, 90%⟩ and ts∈⟨6.5/ωc, 45/ωc⟩,

  • for plants with integration behavior: ηmax∈⟨9.5%, 90%⟩ and ts∈⟨11.5/ωc, 45/ωc⟩,

  • for plants with unstable zero: ηmax∈⟨0%, 90%⟩ and ts∈⟨8.5/ωc, 45/ωc⟩.

A setup for the proposed harmonic excitation based method [7] is in Figure 11, where G(s) is a transfer function of the controlled plant with unknown mathematical model, GR(s) is a PID controller transfer function, and SW is a switch.

Figure 11.

A setup for implementation of the external harmonic excitation based method.

3.1. Process identification using external harmonic excitation

A sinusoidal excitation signal u(t) = Unsin(ωnt) is injected into the plant G(s) when the switch is in the position SW = 4. The plant output y(t) is sinusoidal as well with the same frequency ωn, magnitude Yn and a phase lag φ, that is y(t) = Ynsin(ωnt + φ), where φ = argG(ωn) (Figure 12).

Figure 12.

Time responses of (a) u(t); (b) y(t), and (c) location of G(jωn) in the complex plane.

After obtaining Yn and φ from the recorded time responses u(t) and y(t), one point of the (unknown) plant frequency characteristics related with the excitation frequency ωn can be plotted in the complex plane (Figure 12)

Gjωn=GjωnejargGωn=YnωnUnωneωn.E18

It is recommended to choose Un = (3÷7)%umax [7]. Identified plant parameters are described by the triple {ωn,Yn/Un,φ}. Note that if SW = 4, the identification is performed in open-loop, hence this approach is applicable for stable plants only.

3.2. PID controller tuning rules based on harmonic excitation

Based on identified plant parameters, PID controller can be tuned using the phase margin and/or gain margin approaches. In the control loop in Figure 11, switch SW in ”5” and the PID controller in manual mode. To guarantee a specified phase margin ϕM at the gain crossover frequency ωa*, the closed-loop characteristic equation under a PID controller 1 + L(jω) = 1 + G(jω)GR(jω) = 0 can be easily broken down into the magnitude and phase conditions (ωa* = ωn, and ϕM is the required phase margin, L(jω) is the loop transfer function)

GjωnGRjωn=1,argGωn+argGRωn=180°+φM.E19

To guarantee a specified gain margin GM at the phase crossover frequency ωp*, the closed-loop characteristic equation can be expressed by the magnitude and phase conditions [6] as follows (ωp* = ωn)

GjωnGRjωn=1/GM,argGωn+argGRωn=180°.E20

Graphical interpretation of (19), (20) is shown in Figure 13. Let us denote φ=argGωn, Θ=argGRωn, and consider the ideal PID controller (1a), where K is proportional gain, and Ti, Td are the integral and the derivative time constants, respectively. Substituting for s = jωn into (1a) we obtain

GRjωn=K+jKTdωn1Tiωn.E21

Figure 13.

Graphical interpretation of (a) ϕM, ωa* and shifting G into LA at ωa* = ωn; (b) GM, ωf* and shifting G into LF at ωf* = ωn.

Comparison of (21) with its polar form

GRjωn=GRjωnejΘ=GRjωncosΘ+jsinΘE22

yields a complex Eq. (23) for phase margin approach and (24) for gain margin approach

K+jKTdωn1Tiωn=cosΘGjωn+jsinΘGjωn,E23
K+jKTdωn1Tiωn=cosΘGMGjωn+jsinΘGMGjωn.E24

Finally, PID controller parameters are obtained from (23), (24) using the substitution |GR(jωn)| = 1/|G(jωn)| for the phase margin approach and |GR(jωn)| = 1/[GM|G(jωn)|] for the gain margin approach resulting from (24). The complex equations (23), (24) are solved as a set of two real equations (25) for the phase margin or (26) for the gain margin approaches, respectively

K=cosΘGjωn,KTdωn1βTdωn=sinΘGjωn,E25
K=cosΘGMGjωn,KTdωn1βTdωn=sinΘGMGjωn,E26

where (25a, 26a) represent general rules for calculating the controller gain K. After substituting (25a), (26a) and the ratio β = Ti/Td into (25b), (26b), after some manipulations we obtain a quadratic equation in Td for both approaches

Td2ωn2βTdωnβtgΘ1=0.E27

Expression for calculating Td is the positive solution of (27)

Td=tgΘ2ωn+1ωntg2Θ4+1β.E28

Hence, the PID controller parameters are calculated using the expressions (25a), (26a), Ti = βTd and (28); Θ is obtained from (19b) using (29) for the phase margin approach and (20b) for the gain margin approach

Θ=180°+ϕMargGωn=180°+ϕMφ,E29
Θ=180°argGωn=180°φ.E30

Using the PID controller designed for the phase margin ϕM, the identified point G of the plant Nyquist plot G(jω) with co-ordinates (1) is moved into the point LA of the open-loop Nyquist plot located on the unit circle M1 (Figure 13a). In this way, the gain crossover LA of the open-loop L(jω) is specified

LALjωn=LjωnargLωn=1ϕM,E31

for which the designed PID controller guarantees the required phase margin ϕM; so for ωn is |L(jωn)| = 1. In case of PID controller design for gain margin GM, the identified point G of the plant Nyquist plot G(jω) with co-ordinates (1) is moved into the open-loop frequency response point LF lying on the negative real half-axis of the complex plane (Figure 13b). In this way, the phase crossover LF of the open-loop L(jω) is specified

LFLjωfωn=LjωnargLωn=1GM180°.E32

Location of the points G(jωn) and L(jωn) in the complex plane is shown in Figure 13.

PI, PD and PID tuning formulas for both approaches (ϕM and GM) are summed up in Table 8.

Table 8.

PI, PD and PID controller tuning rules using the harmonic excitation method.

The excitation frequency can be adjusted according to the empirical relations [6].

ωn0.2ωc0.95ωcϕMapproach;ωn0.5ωc1.25ωcGMapproach.E33

3.3. Controller structure selection using the “triangle ruler” rule

The argument Θ in the tuning rules in Table 8 indicates the angle to be contributed to the identified phase φ at ωn by the controller to obtain the resulting open-loop phase (−180°+ ϕM) necessary to guarantee the required phase margin ϕM (or the gain margin GM). Working range of the PID controller argument is given by the union of PI and PD controllers phase ranges

ΘPIDΘPIΘPD=90°0°0°+90°=90°+90°,E34

which is symmetric with respect to 0° and due to frequency properties of PI, PD and PID controllers also upper- and lower-bounded. The working range (34) can be interpreted using a pretended transparent triangular ruler turned according to Figure 14; its segments to the left and right of the axis of symmetry represent the PD and PI working ranges, respectively.

Figure 14.

Controller structure selection using the “triangle ruler“ rule with respect to the situation of (a) G and LA; (b) G and LF.

Figure 14a shows the situation, when the identified point G is situated in the 1st quadrant of the complex plane. In case of phase-margin approach, put this ruler on Figure 14a, the middle of the hypotenuse on the origin of the complex plane and turn it so that its axis of symmetry merges with the ray (0,G). Thus, the ruler determines in the complex plane the cross-hatched area representing the full working range of the PID controller argument. The controller type is chosen depending on the situation of the ray (0,LA) forming with the negative real half-axis the angle ϕM: situation of the ray (0,LA) in the left-hand sector suggests a PD controller, and in the right-hand sector the PI controller. Figure 14a shows the case, when the phase margin ϕM is achievable using both PI or PID controller. According to Figure 14b, the identified point G is placed in 2nd quadrant of the complex plane. Applying the gain-margin approach, the ruler is to be put on Figure 14b according to the similar setup than in case of phase-margin approach. The controller type is chosen depending on the situation of the ray (0,LF) lying on the border of the second and third quadrants of the complex plane; in this case, PI or PID controller type has to be chosen.

3.4. Closed-loop performance

This subsection answers the following question: how to transform the practical performance requirements in terms of maximum overshoot ηmax and settling time ts into the couple of frequency-domain parameters (ωnM) needed for identification and PID controller coefficients tuning?

3.4.1. Systems without integrator

Looking for an appropriate transformation ℜ: (ηmax,ts)→(ωnM) we will consider typical phase margins ϕM given by the set

φMj=20°30°40°50°60°70°80°90°,E35

j = 1…8. Let us split (33a) into 5 equidistant sections Δωn = 0.15ωc, and generate the set of excitation frequencies

ωnk=0.20.350.50.650.80.95ωc=σkωc,E36

k = 1…6. Its elements divided by the plant critical frequency ωc determine the set of so-called excitation levels

σk=0.20.350.50.650.80.95,E37

k = 1…6. Let us demonstrate the qualitative effect of ωnk and ϕMj on closed-loop step response for the plant

G3s=1s+10.5s+10.25s+10.125s+1E38

under PID controllers designed for three phase margins ϕM = 40°, 60°, 80° on three excitation levels σ1 = ωn1c = 0.2; σ3 = ωn3c = 0.5 and σ5 = ωn5c = 0.8. Related closed-loop step responses are shown in Figure 15.

Figure 15.

Closed-loop step responses of G3(s) for various ϕM and ωn.

Achieving required ts and ηmax was tested by designing PID controller for a vast set of benchmark examples [2] at excitation frequencies and phase margins expressed by a Cartesian product ϕMj×ωnk of the sets (35) and (36) for j = 1…8, k = 1…6. Resulting dependences ηmax = f(ϕMn) and ts = (ϕMn) are plotted in Figure 16 [6].

Figure 16.

Dependences: (a) ηmax = f(ϕM,ωn); (b) τs = ωcts = f(ϕM,ωn) for ϕMj×ωnk, j = 1…8, k = 1…6 (relative settling time τs = tsωc).

Considering the frequencies ωa* = ωn are equal which results from the assumptions of the sinusoidal excitation method, the settling time can be expressed by the relation

ts=γπωnE39

similar to (16c) [6], where γ is the curve factor of the step response; in the relation (16c) for the 2nd order closed-loop, γ is from the interval (1;4) and depends on the relative damping [14]. In case of the proposed sinusoid excitation based method γ varies over a considerably broader interval (0.5;16) found empirically and depends strongly on ϕM, that is γ = f(ϕM) at the given excitation frequency ωn. To examine closed-loop settling times for plants with different dynamics, it is advantageous to define the relative settling time [7]

τs=tsωc.E40

Substituting for ωn = σωc into (39) and (40), we obtain a relation for the relative settling time

τs=πσγ,E41

where ts is related to the critical frequency ωc. Due to introducing ωc the right-hand side in (41) is constant for the given plant and independent of ωn. The dependency (41) obtained empirically for different excitation frequencies ωnk is depicted in Figure 16b; it is evident that at every excitation level σk with increasing phase margin ϕM the relative settling time τs first decreases and after achieving its minimum τs_min it increases again. The empirical dependences in Figure 16 have been approximated by quadratic regression curves, thus they are called B-parabolas [7].

3.4.1.1. Discussion

When choosing ϕM = 40° on the B-parabola corresponding to the excitation level σ5 = ωn5c = 0.8 (further denoted as B0.8 parabola), maximum overshoot ηmax = 40% and relative settling time τs≈10 are expected (see Figure 16). Pointcorresponding to these parameters and is located on the left (falling) portion of B0.8 yielding oscillatory step response (see responsein Figure 15c). If the phase margin to ϕM = 60° increases, the relative settling time decreases into the pointon the right (rising) portion of the B0.8 parabola; the corresponding step responsein Figure 15c is weakly aperiodic. For the phase margin ϕM = 80°, the B0.8 parabola indicates a zero maximum overshoot, the relative settling time τs = 20 corresponds to the positionon the B0.8 parabola with aperiodic step response(Figure 15c). If the maximum overshoot ηmax = 20% is acceptable, then ϕM = 53° yields the least possible relative settling time τs = 6.5 on the given level σ5 = 0.8 (“at the bottom” of B0.8).

3.4.1.2. Example 1

Using the sinusoid excitation method, design ideal PID controllers (1a) for an operating amplifier modeled by the transfer function GA(s)

GAs=1TAs+13=1001s+13.E42

The control objective is to guarantee maximum overshoots ηmax1 = 30%, ηmax2 = 5% and a maximum relative settling time τs = 12 in both cases.

3.4.1.3. Solution

  1. Critical frequency of the plant identified by the Rotach test is ωc = 173.216[rad/s] (the process is “fast”). The prescribed settling time is ts = τsc = 12/173.216[s] = 69.3[ms].

  2. For the expected performance (ηmax1s) = (30%;12) (Design No. 1) a satisfactory choice is (ϕM1n1) = (50°;0.5ωc) resulting from the B0.5 parabola in Figure 16. The performance in terms of (ηmax2s) = (5%;12) (Design No. 2) can be achieved by choosing (ϕM2n2) = (70°;0.8ωc) resulting from the B0.8 parabola in Figure 16.

  3. Identified points for the first and second designs are GA(j0.5ωc) = 0.43e−j120° and GA(j0.8ωc) = 0.19e−j165°, respectively. According to Figure 17a, both points are located in the quadrant II of the complex plane, on the Nyquist plot GA(jω) (continuous curve) which verifies the identification.

  4. Using the PID controller designed for (ϕM1n1) = (50°;0.5ωc), the point GA(j0.5ωc) is moved into the gain crossover LA1(j0.5ωc) = 1e−j130° on the unit circle M1, which verifies achieving the phase margin ϕM1 = 180°−130° = 50° (dashed Nyquist plot). The point GA(j0.8ωc) has been moved by the PID controller designed for (ϕM2n2) = (80°;0.8ωc) into LA2(j0.8ωc) = 1e−j110° yielding a phase margin ϕM2 = 180°−110° = 70° (dotted Nyquist plot).

  5. Achieved performance read-off from the closed-loop step response in Figure 17b (dashed line) is ηmax1* = 29.7%, ts1* = 58.4[ms]. Performance in terms of ηmax2* = 4.89%, ts2* = 60.5[ms] identified from the closed-loop step response in Figure 17b (dashed line) complies with the required performance.

Figure 17.

(a) Open-loop Nyquist plots; (b) closed-loop step responses of the operational amplifier, required performance ηmax1 = 30%, ηmax2 = 5% and τs = 12.

3.4.2. Systems with time delay

The sinusoid excitation method is applicable also for plants with time delay commonly considered as difficult-to-control systems [1]. It is a well-known fact that at each frequency ωn∈⟨0,∞) the time delay D turns the phase by ωnD with respect to the delay-free system [44]. For time-delayed systems, the phase condition (29) is extended by an additional phase φD = −ωnD

φ+φ+Θ=180°+ϕM,E43

where φ′ is the phase of the delay-free system and

φ=φ+φDE44

is the identified phase of the plant including the time delay.

The added phase φD = −ωnD is associated with the required phase margin ϕM according to

φ+Θ=180°+ϕM+ωnD.E45

The only modification in using the PID tuning rules in Table 9 is that an increased required phase margin is to be specified.

ϕM=ϕM+ωnDE46

and the controller working angle Θ is to be computed using the relation

Θ=180°φ+ϕM+ωnD.E47
Modelηmaxsωc [rad/s]ts [s]B-par.ϕM/GMωncG(jωn)GR(jωn)ηmax*ts* [s]
GA(s)30%,12173.20.069Figure 1650°0.50.43e−j120°2.31e−j10°29.7%0.058
GA(s)5%,12173.20.069Figure 1670°0.80.19e−j165°5.20ej55°4.89%0.061
GB(s)30%,120.35234.1Figure 1655 + 45.9°0.351.03e−j23°0.97e−j56°18.6%24.78
GB(s)5%,120.35234.1Figure 1670 + 26.2°0.21.09e−j13°0.92e−j71°0.15%28.69
GC(s)30%,200.24183.1Figure 2153 + 10.1°0.3512.7e−j122°0.08ej5.8°29.6%81.73
GC(s)20%,200.24183.1Figure 2162 + 14.5°0.58.10e−j129°0.12e−j28°19.7%82.44
GD(s)30%,120.049245.9Figure 2515 dB1.250.14e−j204°1.47ej24°24.5%241.9
GD(s)5%,120.049245.9Figure 2518 dB0.650.38e−j136°0.38e−j44°4.55%243.4

Table 9.

PID controller design parameters, required and achieved performance, identified plant parameters for GA(s), GB(s), GC(s) and GD(s).

The phase delay ωnD increases with increasing frequency ωn of the sinusoidal excitation signal. It is recommended to use the smallest possible added phase φD = −ωnD to lessen the impact of time delay on closed-loop dynamics.

3.4.2.1. Discussion

The time delay D can be easily specified during identification of the critical frequency as a time D = Ty−Tu, that elapses since the start of the test at time Tu until time Ty, when the system output starts responding to the excitation signal u(t). A small added phase φD = −ωnD due to time delay can be achieved by choosing the smallest possible ωn attenuating the effect of D in (47) and subsequently in the PID controller design. Therefore when designing a PID controller for time delayed systems, it is recommended to choose the lowest possible excitation level when using B-parabolas (most frequently ωnc = 0.2 resp. 0.35) and corresponding couples of B-parabolas in Figure 16. From the given couple (ηmax;ts), ϕM is specified using the chosen couple of B-parabolas, however its increased value ϕM′ given by (46) is to be supplied in the design algorithm thus minimizing effect of the time delay on closed-loop dynamics.

3.4.2.2. Example 2

Using the sinusoid excitation method, design ideal PID controllers (1a) for a distillation column model given by the transfer function GB(s)

GBs=KBeDBsTBs+1=1.11e6,5s3.25s+1.E48

Control objectives are the same as in Example 1.

3.4.2.3. Solution and discussion

  1. Critical frequency of the plant is ωc = 0.3521[rad/s]. Based on comparison of critical frequencies, GB(s) is 500-times slower than GA(s). Required settling time is ts = τsc = 12/0.3521[s] = 34.08[s].

  2. Because DB/TB = 2 > 1, the plant is a so-called “dead-time dominant system.“ Due to a large time delay, it is necessary to choose the lowest possible excitation frequency ωn to minimize the added phase ωnDB in (47). Hence, for the required performance (ηmax2s) = (5%;12) (Design No. 2) we choose the B0.2 parabolas in Figure 16 at the lowest possible level ωnc = 0.2 to find (ϕM2n2) = (70°;0.2ωc). The added phase value is ωn2DB = 0.2ωcDB = 0.2·0.3521·6.5·180/π = 26.2°, hence the phase supplied to the PID design algorithm is ϕ́M2 = ϕM2 + ωn2DB = 70°+ 26.2° = 96.2° (instead of ϕM2 = 70° for a delay-free system). The required performance (ηmax1s) = (30%;12) (Design No. 1) can be achieved by choosing (ϕM1n1) = (55°;0.35ωc) from the B0.35 parabolas in Figure 16 (i.e., ωnc = 0.35). The phase margin ϕ́M1 = 55°+ 45.9° supplied into the design algorithm was increased by ωn1DB = 0.35ωcDB = 0.35·0.3521·6.5·180/π = 45.9° compared with ϕM1 = 55° in case of delay-free system.

  3. Figure 18a shows that identified points GB(j0.35ωc) = 1.03e−j23° and GB(j0.2ωc) = 1.09e−j13° are located in the quadrant I of the complex plane at the beginning of the frequency response GB(jω) (continuous curve).

  4. The point GB(j0.2ωc) (Design No. 2) was shifted by the PID controllers to the open-loop amplitude crossover LB2(j0.2ωc) = 1e−j110° (dotted Nyquist plot in Figure 18a). Note that LB2 has the same position in the complex plane as LA2 in Figure 17a, however at a considerably lower frequency ωn2B = 0.2·0.3521 = 0.07[rad/s] compared to ωn2A = 0.8·173.216 = 138.6[rad/s] (ts2_B* = 28.69[s] is almost 500 times larger than ts2_A* = 0.0584[s] which demonstrates the key role of ωn in achieving required closed-loop dynamics). The identified point GB(j0.35ωc) (Design No. 1) was moved by the designed PID controller into the amplitude crossover LB1(j0.35ωc) = 1e−j125° (dashed Nyquist plot in Figure 18a).

  5. Achieved performances (ηmax1* = 18.6%, ts1* = 24.78[s], dashed line), (ηmax2* = 0.15%, ts2* = 28.69[s], dotted line) in terms of the closed-loop step responses in Figure 18b comply with the required performance specification.

Figure 18.

(a) Open-loop Nyquist plots; (b) closed-loop step responses of the distillation column, required performance ηmax1 = 30%, ηmax2 = 5% and τs = 12.

3.4.3. Systems with 1st order integrator

Corresponding B-parabolas in Figures 1921 were obtained by applying the sinusoid excitation method on a set of benchmark systems with first-order integrator (for a Cartesian product ϕMj×ωnk of the sets (35) and (36), j = 1…8, k = 1…6 and three various ratios Ti/Td: β = 4, 8 and 12).

Figure 19.

B-parabolas: (a) ηmax = f(ϕM,ωn); (b) τs = ωcts = f(ϕM,ωn) for systems with integrator β = 4.

Figure 20.

B-parabolas: (a) ηmax = f(ϕM,ωn); (b) τs = ωcts = f(ϕM,ωn) for systems with integrator β = 8.

Figure 21.

B-parabolas: (a) ηmax = f(ϕM,ωn); (b) τs = ωcts = f(ϕM,ωn) for systems with integrator β = 12.

3.4.3.1. Discussion

Inspection of Figures 19a, 20a and 21a reveals that increasing β results in decreasing of the maximum overshoot ηmax, narrowing of the B-parabolas of relative settling times τs = f(ϕMn) for each identification level ωnc and consequently increasing the settling time.

Consider for example, the B0.95 parabolas in Figures 19b, 20b and 21b: if ϕM = 70° and β = 4 the relative settling time is τs = 30, for β = 8 it grows up to τs = 40, and for β = 12 even to τs = 45. If a 10% maximum overshoot is acceptable for the given system with integrator, then the standard interaction PID controller can be used with no need to use the setpoint filter; however a larger settling time is expected.

3.4.3.2. Example 3

Using the sinusoidal excitation method, let us design ideal PID controllers for a flow valve modeled by the transfer function GC(s) (system with an integrator and a time delay)

GCs=KCeDCssTCs+1=1.3e2.1ss7.51s+1.E49

The control objective is to guarantee a maximum overshoot of the closed-loop step response ηmax1 = 30%, ηmax2 = 20% and a maximum relative settling time τs = 20.

3.4.3.3. Solution and discussion

  1. Critical frequency of the plant identified by the Rotach test is ωc = 0.2407[rad/s]. Then, the required settling time is ts = τsc = 20/0.2407[s] = 83.09[s].

  2. For GC(s) the time delay/time constant ratio is DC/TC = 2.1/7.51 = 0.28 < 1, hence, the influence of the time constant prevails—GC(s) is a so-called “lag-dominant system” with integrator, therefore B-parabolas are to be chosen carefully. From one side, due to time delay it would be desirable to choose B-parabolas from Figures 19, 20 or 21 with the lowest identification level ωnc = 0.2. However, the minima of B0.2 parabolas in Figure 19b (for β = 4), Figure 20b (for β = 8) and Figure 21b (for β = 12) indicate that the smallest feasible relative settling time τs = 36.5 (for β = 4), τs = 33 (for β = 8) and τs = 34 (for β = 12), which do not satisfy the required value τs = 20.

  3. The first performance specification (ηmax1s) = (30%;20) can be provided using the B0.35 parabolas for β = 12 (Figure 21b) at the level ωnc = 0.35 and for parameters (ϕM1n1) = (53°;0.35ωc) (Design No. 1), supplying the augmented open-loop phase margin ϕ´M1 = ϕM1 + ωn1DC = 53° + 10.1° = 63.1° into the PID controller design algorithm. The second performance specification (ηmax2s) = (20%,20) can be achieved using the B0.5 parabolas in Figure 21 for β = 12 and ωnc = 0.5 and parameters (ϕM2n2) = (62°;0.5ωc) (Design No. 2). To reject the influence of DC, instead of ϕM2 = 62° the augmented open-loop phase margin ϕ´M2 = ϕM2 + ωn1DC = 62° + 14.5°= 76.5° was supplied into the PID controller design algorithm.

  4. Identified points GC(j0.35ωc) = 12.7e−j122° and GC(j0.5ωc) = 8.10e−j129° are located on the plant frequency response GC(jω) (continuous curve) in Figure 22a verifying correctness of the identification.

  5. Using the PID controller, the first identified point GC(j0.35ωc) (Design No. 1) was moved into the gain crossover LC1(j0.35ωc) = 1e−j127° located on the unit circle M1; this verifies achieving the phase margin ϕM1 = 180°−127° = 53° (dashed Nyquist plot in Figure 22a). Achieved performance in terms of the closed-loop step response in Figure 22b is ηmax1* = 29.6%, ts1* = 81.73[s] (dashed line).

  6. The second identified point GC(j0.5ωc) (Design No. 2) was moved into LC2(j0.5ωc) = 1e−j118° achieving the phase margin ϕM2 = 180°−118° = 62° (dotted Nyquist plot in Figure 22a). Achieved performance in terms of the closed-loop step response parameters (Figure 22b) ηmax2* = 19.7%, ts2* = 82.44[s] (dotted line) meets the required specification. Frequency characteristics LC1(jω), LC2(jω) begin near the negative real half-axis of the complex plane because both open-loops contain a 2nd order integrator.

Figure 22.

(a) Open-loop Nyquist plots; (b) closed-loop step responses of the flow valve, required performance ηmax1 = 30%, ηmax2 = 20% and τs = 20.

3.4.4. Systems with unstable zero

Consider typical gain margins GM given by the set

GMj=3dB5dB7dB9dB11dB13dB15dB17dBE50

for j = 1…8. Let us split (33b) into five equal sections and generate the set of excitation frequencies

ωnk=0.5ωc0.65ωc0.8ωc0.95ωc1.1ωc1.25ωcE51

for k = 1…6. Its elements divided by the plant critical frequency ωc determine excitation levels

σk=ωnk/ωcσk=0.50.650.80.951.11.25E52

for k = 1…6. Figure 23 shows closed-loop step response shaping using different GM and ωn in the PID tuning for the plant (53b) with parameters T3 = 0.75, α3 = 1.3, for four required gain margins GM = 5 dB, 9 dB, 11 dB and 13 dB, and three different excitation levels σ1 = ωn1c = 0.5, σ3 = ωn3c = 0.8 and σ5 = ωn5c = 1.1.

Figure 23.

Closed-loop step responses of the plant G3(s) under PID controllers designed for various GM and ωn.

Consider the following benchmark plants

G2s=α2s+1T2s+1n2,G3s=α3s+1s+1T3s+1T32s+1T33s+1.E53

The proposed method has been applied for each element of the Cartesian product ωnk × GMj of the sets (51) and (50). Significant differences between dynamics of individual control loops under designed PID controllers can be observed for the benchmark systems (53).

Consider the benchmark plants G2(s) and G3(s) with following parameters: G2.1(s): (T2,n22) = (0.75,8,0.2); G2.2(s): (1,3,0.1); G2.3(s): (0.5,5,1); G3(s): T3 = 0.5, α3 = 1.3.

Couples of examined plants [G3(s), G2.3(s)] and [G2.2(s), G2.1(s)] differ principally by the ratio α/T, which is significant for the closed-loop performance assessment for plants with an unstable zero (for the 1st couple [α3/T3 = 2.6, α2.3/T2.3 = 2], for the 2nd couple [α2.2/T2.2 = 0.1, α2.1/T2.1 = 0.27]).

According to the ratio α/T unknown plants with an unstable zero can be classified in following two groups [7]:

  1. plants with α/T < 0.3;

  2. plants with α/T > 0.3.

With respect to this classification, B-parabolas ηmax = f(GMn), τs = f(GMn) for nonminimum phase systems with an unstable zero constructed for different open-loop gain margins GM and excitation levels σ are depicted in Figure 24 (for α/T > 0.3) and in Figure 25 (for α/T < 0.3).

Figure 24.

B-parabolas: (a) ηmax = f(GM,ωn); (b) τs = ωcts = f(GM,ωn) for nonminimum phase systems, α/T > 0.3.

Figure 25.

B-parabolas: (a) ηmax = f(GM,ωn); (b) τs = ωcts = f(GM,ωn) for nonminimum phase systems, α/T < 0.3.

3.4.4.1. Example 4

Using the sinusoid excitation method, ideal PID controllers are to be designed for a heating plant described by the transfer function GD(s) (a system with an unstable zero)

GDs=KDTzs+1TDs+13=0.87.5s+127.5s+13.E54

The control objective is to guarantee a maximum overshoot ηmax1 = 30%, ηmax2 = 5% and maximum relative settling time τs = 12.

3.4.4.2. Solution and discussion

  1. Critical frequency of the plant identified by the Rotach test is ωc = 0.0467[rad/s], the system is ”slow“. The required settling time is ts = τsc = 12/0.0488 = 245.90[s].

  2. Because α/TD = 7.5/27.5 = 0.27 < 0.3, the gain margin GM and the excitation frequency ωn of the controlled object GD(s) will be determined using B-parabolas in Figure 25. For the required performance (ηmax1s) = (30%,12) the appropriate values of gain margin and excitation frequency are (GM1n1) = (15 dB,1.25ωc), that is “gray parabolas” in Figure 25. Similarly, the performance (ηmax2s) = (5%,12) can be achieved by choosing (GM2n2) = (18 dB,0.65ωc) according to “violet” B-parabolas in Figure 25.

  3. Examination of the Nyquist plots of the controlled object GD(jω) and the open-loops LD1(jω), LD2(jω) in Figure 26a reveals that the first identified point GD(j1.25ωc) is located in the quadrant III of the complex plane, and its identification is carried out under a relatively low frequency 1.25ωc = 1.25·0.0467 = 0.0584[rad/s], hence no high-frequency noise corrupting the excitation and output signals u(t) and y(t), respectively, is expected during identification. If, however, the identification at the excitation level ωn = 1.25ωc were carried out for a “fast” object with a high value of ωc, it would be necessary to choose the lowest possible excitation level in order to reject the identification noise. The second identified point GD(j0.65ωc) is placed in the quadrant II of the complex plane.

  4. Using the PID controller designed for (GM1n1) = (15 dB,1.25ωc) the point GD(j1.25ωc) was compensated into the target point LD1j1.25ωc=1/10GM1/20ej180°located on the negative real half-axis where the gain margin GM1 of the open-loop LD1(jω) (red Nyquist plot) is satisfied. The achieved performance evaluated from the closed-loop step response in Figure 26b is ηmax1* = 24.5%, ts1* = 241.88[s].

  5. Using the PID controller designed for (GM2n2) = (18 dB,0.65ωc), the point GD(j0.65ωc) was moved to the target point LD2j0.65ωc=1/10GM2/20ej180°where the gain margin GM2 of the open-loop LD2(jω) (green Nyquist plot) is satisfied. The achieved performance ηmax2* = 4.55%, ts2* = 243.42[s] evaluated from the closed-loop step response in Figure 26b satisfies the control objective.

Figure 26.

(a) Open-loop Nyquist plots; (b) closed-loop step responses of the heating system, required performance ηmax1 = 30%, ηmax2 = 5% and τs = 12.

Time-domain performance requirements specified by the process technologist, identified plant parameters needed for PID controller tuning (for two PID controllers of all four plants GA(s), GB(s), GC(s) and GD(s)) along with specified and achieved performance measure values are summarized in Table 9. The asterisk “*“ indicates closed-loop performance complying with the required one.

3.5. Robust PID controller design for specified performance

When identifying an uncertain plant, the sinusoidal excitation with the frequency ωn is repeated for individual parameter changes to obtain a set of points Gi from the set of frequency responses of the uncertain plant

Gijωn=GijωnejargGiωn=ai+jbi,E55

i = 1,2…N. Plant parameter changes are reflected in changes of the magnitude and the phase |Gi(jωn)| and argGin), respectively; i = 1…N; N = 2p is the number of identification experiments and p is the number of varying technological quantities of the plant. The nominal model G0(jωn) is obtained from mean values of the real and imaginary parts of Gi(jωn)

G0jωn=a0+jb0=1Ni=1Nai+j1Ni=1Nbi,E56

i = 1,2…N. Obviously

G0jωn=a02+b02=1Ni=1Nai2+i=1Nbi2;argG0jωn=arctgb0a0=arctgi=1Nbii=1Nai,E57

where φ0n) = arg{G0(jωn)}. The points Gi represent some elements of the family of plants and can be enclosed by a circle MG centered in G0(jωn) with the radius RG≡RGn) corresponding to the maximum distance between Gi(jωn) and G0(jωn)

RG=maxiaia02+bib02,E58

i = 1,2…N. Actually, the control law generated by the robust controller GRrob(s) designed for the nominal point G0(jωn) performs the mapping

:RGRL:RL=GRrobRGE59

of the set of identified points Gi(jωn) encircled by MG with the radius RG onto the set of points Li(jωn) delineated by ML and calculates the radius RL≡RLn) of the dispersion circle ML which encloses the points Li(jωn) of the Nyquist plot so as to guarantee fulfillment of the robust performance condition.

A robust PID controller is designed using the sinusoidal excitation method with input data for the nominal model G0(jωn): {|G0(jωn)|; φ0 = argG0n)}. Substituting them into (25a), (26a), (29) and (30), the expressions for calculating robust PID controller parameters according to Table 8 are obtained. Obviously, the phase and gain margins ϕM and GM, respectively, are robust PID controller tuning parameters and at the same time attractive robustness measures [6].

3.5.1. Robust performance condition

Theorem 1 (Sufficient condition for robust performance under a PID controller).

Consider an uncertain continuous-time stable dynamic system described by a nominal model and unstructured uncertainty. The PID controller GR(s) tuned according to the rules in Table 8 guarantees robust closed-loop performance if the following conditions are satisfied

φM>arccos112χLRGωnG0jωn+χSsinφS2,GM>1+χLRGωnG0jωn1χSGS1GS,E60

where ϕM and GM are the required phase and gain margins, respectively, ωn is the excitation frequency, χL = RL+/RL and χS = RS+/RS are safety factors of radii of the dispersion circles ML and MS, respectively, delineating prohibited areas; RGn) is the radius of the dispersion circle at the Nyquist plot of the plant at ωn, and G0n) is a point at the Nyquist plot of the nominal plant at ωn. The prohibited area MS can be defined in terms of ϕM or GM using the expressions ϕS = arcsin(RS) or GS = 1/(1−RS), respectively.

Proof:

The proof is straightforward using Figures 27 and 28. If the nominal open-loop L0(s) = G0(s)GR(s) is stable, then according to the Nyquist stability criterion the closed-loop with the uncertain plant will be stable if the distance between L0 and (−1, j0), that is |1 + L0(jωn)| is greater than the sum of the radii RLn) of the circle ML centered in L0, and RSn) of the circle MS centered in (−1.0j), that is

RL+RS<1+L0jωn.E61

Figure 27.

Dispersion circles MG and ML and the prohibited area delineated by the circle MS for the phase-margin approach.

Figure 28.

Dispersion circles MG and ML and the prohibited area delineated by the circle MS for gain-margin approach.

3.5.1.1. Robust performance condition: phase margin approach

According to Figure 27, the distance between (−1.0j) and the open-loop Nyquist plot L0(jωn) at ωn, that is |1 + L0(jωn)| can be calculated by applying the cosine rule to the triangle given by the vertices (−1.0,L0) (ϕM is the phase margin)

1+L0=1+L022L0cosφM.E62

From the principles of the sinusoidal excitation PID controller tuning method results that the robust controller shifts the nominal point of the plant frequency response G0 to the point L0 situated on the unit circle. Thus, ωn becomes the gain crossover frequency. As the point L0 is situated on the unit circle M1, the magnitude |L0(jωn)| equals one, that is |L0| = |G0||GR| = 1, yielding the transformation ratio |GR| = |G0|−1 between the radii RG and RL of the circles MG and ML, respectively. The radius RL of the dispersion circle ML can be expressed as

RL=RGGR=RGG0.E63

Substituting (62) and (63) in (61) yields the robust performance condition

χLRGωnG0jωn+χSRS2<22cosφM,E64

which after some manipulations is identical to the proven condition (60a). Typical values of safety factors are χL = 1.1 and χS = 1.2. The value of ϕM chosen according to the robust performance condition used in the tuning rules in Table 1 yields robust PID controller coefficients. The design procedure is illustrated in Section 3.5.1.3.

3.5.1.2. Robust performance condition: gain margin approach

According to Figure 28, |1 + L0(jωn)| is a complement of |0,L0| = |L0| to the unit value. Thus

L0jωn+1+L0jωn=11+L0jωn=1L0jωn.E65

From the principles of the proposed PID tuning method results that the robust controller shifts the point G0n) of the plant nominal frequency response to L0 situated on the negative real half-axis of the complex plane. From the relation |L0(jωn)| = |G0(jωn)||GR(jωn)| = 1/GM results the ratio |GR(jωn)| = 1/[GM|G0(jωn)|] between the radii RG and RL = |GR|RG of the circles MG and ML, respectively. The radius RL of the dispersion circle ML is calculated as follows

RL=RGGR=RGGMG0jωn.E66

Substituting (65) and (66) into the general robust performance condition (61) and considering the safety factors χL and χS, the following inequality is obtained

GM1GM>χLRGGMG0jωn+χSRS,E67

which after some manipulations is identical to the proven condition (60b). According to the robust performance condition, the chosen value GM is substituted into (26a) and robust PID controller parameters are obtained from Table 8. ϕS and GS are found from the B-parabolas (Figures 16, 1921, 24, 25) considering ηmaxN and τsN of the worst-case plant.

3.5.1.3. Examples

3.5.1.3.1. Example 5

Consider the plant model GA(s) from Subsection 3.4

GA0s=KA0TA0s+13=10.01s+13E68

to be the nominal model of an uncertain system where KA and TA are uncertain parameters varying within ±15% from their nominal values KA0 and TA0 (i.e., the total dispersion is κ = 30%). Let us design a robust PID controller to guarantee ηmaxN = 30% and a relative settling time τsN = 12 for the worst-case model of GA(s).

3.5.1.3.2. Robust PID controller design for the uncertain plant GA(s)—solution and discussion

  1. The measured ultimate frequency of the nominal model is ωc0 = 173.216[rad/s]. From the robust performance condition results tsN = τsNc = 12/173.216 = 69.3[ms].

  2. For the required performance (ηmaxNsN) = (30%,12) the corresponding values of phase margin and excitation frequency have been selected (ϕMn0) = (50°,0.5ωc0) using the pair of ”red“ B-parabolas in Figure 16. As there are two uncertainties in GA(s) (KA and TA), the number of identification experiments is N = 22 = 4.

  3. For ωn0 = 0.5,ωc0 = 0.5·173.21 = 86.61[rad/s], four points of the family of Nyquist plots corresponding to the uncertain plant model were identified using the sinusoidal excitation: GA1(jωn0), GA2(jωn0), GA3(jωn0) and GA4(jωn0) (blue ”x “in Figure 29a). The nominal point GA0(jωn0) calculated from the coordinates of all identified points GAi(jωn0), i = 1, 2, 3, 4 is located on the “blue” Nyquist plot of the nominal model GA0(jωn). The dispersion circle MG is centered in GA0(jωn0) with the radius RG = 0.199.

  4. The desired robust performance ηmaxN = 30%, τsN = 12 can be achieved using ϕS = 50° at the excitation level ωn0 = 0.5ωc0.

  5. R.H.S. of the robust performance condition (60a) is δ0_RP = 89.39°, thus the robust performance condition (60a) ϕM0_RP will be satisfied if choosing for example ϕM = 90°.

  6. Using the designed PID controller, the nominal point GA0(jωn0) is shifted to LA0(jωn0) = GA0(jωn0)GR_rob(jωn0) = 1e−j90° on the unit circle M1 (Figure 29a).

  7. The smallest phase margin estimated from the location of the worst-case point LAN = 1.13e−j114° is ϕMN = 66.2°. The achieved smallest phase margin ϕ+MN = 61° is given by the intersection of the “red” Nyquist plot and the unit circle M1 (closest to the negative real half-axis).

  8. Radius of the prohibited area RS = sinϕS = sin(50·3.14/180) = 0.766/χS = 0.6383 multiplied by the expansion coefficient χS = 1.2, as well as the χS = 1.1-times enlarged radius RL of the dispersion circle ML guarantee that none of the open-loop Nyquist plots enters the prohibited area delineated by the MS circle. The enlarged circles ML+ and MS+ (dotted plots in Figure 29a) are touching, which indicates fulfillment of the robust performance condition.

  9. From the closed-loop step response of the worst-case plant model (Figure 29b, red plot) results ηmaxN_obtained = 8.2% and the relative settling time τsN_obtained = ωc0tsN_obtained = 173.216·0.0671 = 11.62, which proves achievement of the specified performance. Using the phase margin ϕM = 90° at the identification level ωn0 = 0.5ωc0 (“red” B-parabolas in Figure 16a) corresponds to the nominal performance ηmax0 = 2% and τs0 = 10. The closed-loop step response in Figure 29b (green plot) corresponding to the nominal model satisfies ηmax0_obtained = 0% and τs0_obtained = 173.216·0.0469 = 8.12 as expected.

Figure 29.

(a) Nyquist plots for GA(s), ηmaxN = 30% and τsN = 12; (b) closed-loop step responses satisfy the required performance ηmaxN = 30% and τsN = 12 (upper plot: worst-case plant model; lower plot: nominal plant model).

3.5.1.3.3. Example 6

Consider the plant model from the Subsection 3.4

GD0s=KD0αDs+1TD0s+13=0.87.5s+127.5s+13E69

to be the nominal model of the uncertain plant GD(s) with parameters KD, TD and αD varying within ±15% from their nominal values KS0, TS0 and αD0 (the total dispersion is κ = 30%). A robust PID controller is to be designed to guarantee specified performance in terms of a maximum overshoot ηmaxN = 5% and a relative settling time τsN = 12 for the worst-case model of GD(s).

3.5.1.3.4. Robust PID controller design for the uncertain plant GD(s)—solution and discussion

  1. The measured ultimate frequency of the nominal model is ωc0 = 0.0488[rad/s]. From the requirements on the nominal closed-loop performance results: ts = τs0c = 12/0.0488 = 245.9[s].

  2. For the required performance (ηmaxNsN) = (5%,12) the corresponding values of gain margin and excitation frequency have been selected (GMn) = (18 dB,0.65ωc0) using the pair of ”red“ B-parabolas in Figure 25. As there are three uncertain parameters in GD(s) (KD, TD and αD), the number of identification experiments is N = 23 = 8.

  3. For ωn0 = 0.65ωc0 = 0.65·0.04880 = 0.03172[rad/s], eight points of the family of Nyquist plots corresponding to the uncertain plant model were identified using the sinusoidal excitation: GD1(jωn)…GD8(jωn) (depicted by blue ”x“ in Figure 30). The nominal point GD0(jωn) calculated from the coordinates of all identified points GDi(jωn), i = 1…8 is located on the Nyquist plot of the nominal model GD0(jωn) (blue curve) thus proving correctness of the identification. Radius of the dispersion circle MG centered in the nominal point GD0(jωn0) with the radius RG = 0.164.

  4. The desired robust performance ηmaxN = 30%, τsN = 12 can be achieved using ϕS = 50° at the excitation level ωn0 = 0.5ωc0.

  5. Using the designed robust PID controller, the nominal point GD0(jωn0) of the plant is shifted to the point LD0(jωn0) = GD0(jωn0)GR_rob(jωn0) = 0.0841e−j180° located on the unit circle. The nominal open-loop Nyquist plot (green plot) crosses LD0(jωn0) (Figure 14), the radius of the circle ML is RL = 0.0400.

  6. The smallest gain margin G+MN = 18.8 dB is estimated from the position of the worst-case point LDN(jωn0) = 0.112e−j197°. The achieved smallest gain margin is given by the intersection point of the red Nyquist plot with the negative real half-axis of the complex plane G+MN = 16.9 dB.

  7. Both the radius of the prohibited area RS = (GS−1)/GS = (1018/20–1)/1018/20 = 0.8741/χS = 0.699 multiplied by the expansion coefficient χS = 1.2, as well as the radius RL of the dispersion circle ML enlarged χS = 1.1-times guarantee that none of the open-loop Nyquist plots enters the prohibited area delineated by the MS circle. The enlarged circles ML+ a MS+ in Figure 30a (dotted curves) touch, which indicates fulfillment of the robust performance condition.

  8. As GM = 21.5 dB at the excitation level ωn0 = 0.65ωc0 has been considered, according to ”pink “B-parabolas in Figure 25a nominal performance ηmax0 = 1.5% and τs0 = 21 is expected. The nominal closed-loop step response in Figure 30b (green plot) shows the nominal performance in terms of ηmax0_obtained = 0% and τs0_obtained = 0.0488·381 = 18.59 as expected.

  9. From the closed-loop step response of the worst-case plant model (Figure 30b, red plot) results ηmaxN_obtained = 4.8% and the relative settling time τsN_obtained = ωc0tsN_obtained = 0.0488·237 = 11.57 which proves achievement of the specified performance. Using the gain margin GM = 21.5 dB at the excitation level ωn0 = 0.65ωc0 (“pink“ B-parabolas in Figure 25a) indicates the expected nominal performance ηmax0 = 1.5% and τs0 = 21. The closed-loop step response in Figure 30b (green plot) corresponding to the nominal model satisfies ηmax0_obtained = 0% and τs0_obtained = 0.0488·381 = 18.59 as expected.

Figure 30.

(a) Nyquist plots for GD(s), ηmaxN = 5% and τsN = 12; (b) closed-loop step responses with GD(s) for the required performance ηmaxN = 5% and τsN = 12 (upper plot: worst-case plant model; lower plot: nominal plant model).

4. Conclusion

A novel frequency-domain PID design method for performance specified in terms of maximum overshoot and settling time is presented applicable for uncertain systems with parametric uncertainties. One of the main results is developed empirical charts called B-parabolas; this insightful graphical tool is used to transform engineering time-domain performance specifications (maximum overshoot and settling time) into frequency-domain performance measures (phase margin and gain margin). The developed PID design method is based on shaping the closed-loop step response using various combinations of excitation signal frequencies and required phase and gain margins. Using B-parabolas, it is possible to shape time responses of processes with various types of dynamics. By applying appropriate PID controller design methods including the above presented, it is possible to achieve cost-effective control of processes with uncertainties. The presented advanced external harmonic excitation-based design method contributes to improve the unfavorable statistical ratio between the properly tuned to all implemented PID controllers in industrial control loops.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Štefan Bucz and Alena Kozáková (September 12th 2018). Advanced Methods of PID Controller Tuning for Specified Performance, PID Control for Industrial Processes, Mohammad Shamsuzzoha, IntechOpen, DOI: 10.5772/intechopen.76069. Available from:

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