Li, Y. and Ang, K.H. and Chong, G.C.Y. (2006) PID control system
`analysis and design. IEEE Control Systems Magazine 26(1):pp. 32-41.
`
`http://eprints.gla.ac.uk/3815/
`
`Deposited on: 13 November 2007
`
`Glasgow ePrints Service
`http://eprints.gla.ac.uk
`
`BTL EX2015
`Allergan v. BTL
`PGR2021-00024
`
`

`

`PID Control System
`PID Control System
`Analysis and Design
`Analysis and Design
`
`PROBLEMS, REMEDIES, AND FUTURE DIRECTIONS
`
`By YUN LI, KIAM HEONG ANG, and GREGORY C.Y. CHONG
`
`©IMAGESTATE
`
`W ith its three-term functionality offering treatment of both transient and steady-state responses,
`
`proportional-integral-derivative (PID) control provides a generic and efficient solution to real-
`world control problems [1]–[4]. The wide application of PID control has stimulated and sus-
`tained research and development to “get the best out of PID’’ [5], and “the search is on to find
`the next key technology or methodology for PID tuning” [6].
`This article presents remedies for problems involving the integral and derivative terms. PID design objec-
`tives, methods, and future directions are discussed. Subsequently, a computerized, simulation-based approach
`is presented, together with illustrative design results for first-order, higher order, and nonlinear plants. Finally,
`we discuss differences between academic research and industrial practice, so as to motivate new research
`directions in PID control.
`
`STANDARD STRUCTURES OF PID CONTROLLERS
`
`Parallel Structure and Three-Term Functionality
`The transfer function of a PID controller is often expressed in the ideal form
`
`32 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2006
`
`1066-033X/06/$20.00©2006IEEE
`
`

`

`where GPD(s) and GPI(s) are the factored PD and PI parts of
`the PID controller, respectively, and
`α = 1 ± √
`1 − 4TD/TI
`2
`
`> 0.
`
`THE INTEGRAL TERM
`
`Destabilizing Effect of the Integral Term
`Referring to (1) for TI (cid:4)= 0 and TD = 0, it can be seen that
`adding an integral term to a pure proportional term increases
`the gain by a factor of
`
`(cid:4)(cid:4)(cid:4)(cid:4)1 + 1
`
`jωTI
`
`(cid:4)(cid:4)(cid:4)(cid:4) =
`
`(cid:5)
`1 + 1
`ω2T2
`I
`
`> 1, for all ω,
`
`(5)
`
`and simultaneously increases the phase-lag since
`
`(cid:3)
`
`(cid:2)
`1 + 1
`jωTI
`
`(cid:3)
`
`(cid:2) −1
`
`ωTI
`
`= tan
`−1
`
`< 0, for all ω.
`
`(6)
`
`Hence, both gain margin (GM) and phase margin (PM) are
`reduced, and the closed-loop system becomes more oscillatory
`and potentially unstable.
`
`Integrator Windup and Remedies
`If the actuator that realizes the control action has saturated
`range limits, and the saturations are neglected in a linear con-
`trol design, the integrator may suffer from windup; this caus-
`es low-frequency oscillations and leads to instability. The
`windup is due to the controller states becoming inconsistent
`with the saturated control signal, and future correction is
`ignored until the actuator desaturates.
`
`Automatic Reset
`If TI ≥ 4TD so that the series form (3) exists, antiwindup can be
`achieved implicitly through automatic reset. The factored PI
`part of (3) is thus implemented as shown in Figure 1 [8], [9].
`
`Explicit Antiwindup
`In nearly all commercial PID software packages and hardware
`modules, however, antiwindup is implemented explicitly
`through internal negative feedback, reducing UI(s) to [8]–[10]
`
`GPID(s) = U(s)
`E(s)
`
`= KP
`
`(cid:2)
`1 + 1
`TIs
`
`(cid:3)
`+ TDs
`
`,
`
`(1)
`
`where U(s) is the control signal acting on the error signal E(s),
`KP is the proportional gain, TI is the integral time constant, TD
`is the derivative time constant, and s is the argument of the
`Laplace transform. The control signal can also be expressed in
`three terms as
`
`(2)
`
`U(s) = KPE(s) + KI
`E(s) + KDsE(s)
`= UP(s) + UI(s) + UD(s),
`
`1 s
`
`where KI = KP/TI is the integral gain and KD = KPTD is the
`derivative gain. The three-term functionalities include:
`1) The proportional term provides an overall control action
`proportional to the error signal through the allpass gain
`factor.
`2) The integral term reduces steady-state errors through
`low-frequency compensation.
`3) The derivative term improves transient response through
`high-frequency compensation.
`A PID controller can be considered as an extreme form of a
`phase lead-lag compensator with one pole at the origin and
`the other at infinity. Similarly, its cousins, the PI and the PD
`controllers, can also be regarded as extreme forms of phase-
`lag and phase-lead compensators, respectively. However, the
`message that the derivative term improves transient response
`and stability is often wrongly expounded. Practitioners have
`found that the derivative term can degrade stability when
`there exists a transport delay [4], [7]. Frustration in tuning KD
`has thus made many practitioners switch off the derivative
`term. This matter has now reached a point that requires clarifi-
`cation, as discussed in this article. For optimum performance,
`KP, KI (or TI), and KD (or TD) must be tuned jointly, although
`the individual effects of these three parameters on the closed-
`loop performance of stable plants are summarized in Table 1.
`
`The Series Structure
`If TI ≥ 4TD, the PID controller can also be realized in a series
`(cid:2)
`(cid:3)
`form [7]
`1 + 1
`αTIs
`
`(3)
`
`GPID(s) = (α + TDs) KP
`= GPD(s)GPI(s),
`
`(4)
`
`TABLE 1 Effects of independent P, I, and D tuning on closed-loop response.
`For example, while K I and K D are fixed, increasing K P alone can decrease rise time,
`increase overshoot, slightly increase settling time, decrease the steady-state error, and decrease stability margins.
`
`Increasing K P
`Increasing K I
`Increasing K D
`
`Rise Time
`Decrease
`Small Decrease
`Small Decrease
`
`Overshoot
`Increase
`Increase
`Decrease
`
`Settling Time
`Small Increase
`Increase
`Decrease
`
`Steady-State Error
`Decrease
`Large Decrease
`Minor Change
`
`Stability
`Degrade
`Degrade
`Improve
`
`FEBRUARY 2006 « IEEE CONTROL SYSTEMS MAGAZINE 33
`
`(cid:4)
`

`

`UPD(s)
`
`KP
`
`U(s)
`
`++
`
`Actuator Model
`
`U(s)
`
`1
`1 + αTIs
`
`FIGURE 1 The PI part of a PID controller in series form for automatic
`reset. The PI part of a PD-PI factored PID transfer function can be
`configured to counter actuator saturation without the need for sepa-
`rate antiwindup action. Here, UPD(s) is the control signal from the
`preceding PD section. When U(s) does not saturate, the feedfor-
`ward-path gain is unity and the overall transfer function from UPD(s)
`to ¯U(s) is thus 1 + 1/(αTIs), the same as the last factor of (3).
`(cid:6)
`(cid:7)
`KPE(s) − U(s) − ¯U(s)
`
`,
`
`γ
`
`(7)
`
`˜UI(s) = 1
`TIs
`
`where U(s) is the theoretically computed control signal, ¯U(s)
`is the actual control signal capped by the actuator limits, and
`γ is a correcting factor. A value of γ in the range [0.1, 1.0] usu-
`ally results in satisfactory performance when the PID coeffi-
`cients are reasonably tuned [7].
`
`Accounting for Windup in Design Simulations
`Another solution to antiwindup is to reduce the possibilities
`for saturation by reducing the control signal, as in linear
`quadratic optimal control schemes that minimize the tracking
`error and control signal through a weighted objective func-
`tion. However, preemptive minimization of the control signal
`can impede performance due to minimal control amplitude.
`Therefore, during the design evaluation and optimization
`process, the control signal should not be minimized but
`rather capped at the actuator limits when the input to the
`plant saturates since a simulation-based optimization process
`can automatically account for windup that might occur.
`
`THE DERIVATIVE TERM
`
`Stabilizing and Destabilizing Effect of the Derivative Term
`Derivative action is useful for providing phase lead, which
`offsets phase lag caused by integration. This action is also
`helpful in hastening loop recovery from disturbances. Deriva-
`tive action can have a more dramatic effect on second-order
`plants than first-order plants [9].
`However, the derivative term is often misunderstood and
`misused. For example, it has been widely perceived in the
`control community that the derivative term improves tran-
`sient performance and stability. But this perception is not
`always valid. To see this, note that adding a derivative term to
`a pure proportional term reduces the phase lag by
`
`(cid:8)
`1 + jωTD
`
`(cid:9) = tan
`
`−1
`
`∈ [0, π/2] for all ω,
`
`(8)
`
`ωTD
`1
`
`34 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2006
`
`which tends to increase the PM. In the meantime, however,
`(cid:10)
`the gain increases by a factor of
`1 + ω2T2
`
`> 1, for all ω,
`
`D
`
`(9)
`
`(cid:4)(cid:4)1 + jωTD
`
`(cid:4)(cid:4) =
`
`and hence the overall stability may be improved or degraded.
`To demonstrate that adding a differentiator can destabilize
`some systems, consider the typical first-order delayed plant
`G(s) = K
`1 + Ts
`where K is the process gain, T is the time constant, and L is the
`dead time or transport delay. Suppose that this plant is con-
`trolled by a proportional controller with gain KP and that a
`derivative term is added. The resulting PD controller
`GPD(s) = KP(1+TDs)
`
`(10)
`
`(11)
`
`−Ls,
`e
`
`leads to an open-loop feedforward path transfer function with
`frequency response
`G( jω)GPD( jω) = KKP
`(cid:5)
`
`For all ω, the gain satisfies
`1 + T2
`1 + T2ω2
`where inequality (13) holds since ((1 + T2
`monotonic in ω.
`Hence, if KP > 1/K and TD > T/K KP, then, for all ω,
`
`KKP
`
`ω2
`
`D
`
`(cid:2) KKP min
`
`D
`
`(cid:4)(cid:4)G( jω)GPD( jω)
`(cid:4)(cid:4) > 1.
`
`1 + jTDω
`− jLω.
`1 + jTω e
`(cid:2)
`1, TD
`T
`ω2)/(1 + T2ω2))1/2 is
`
`(cid:3)
`
`,
`
`(12)
`
`(13)
`
`(14)
`
`Inequality (14) implies that the 0-dB gain crossover frequency is
`at infinity. Furthermore, due to the transport delay, the phase is
`(cid:4) G( jω)GPD( jω) = tan
`− tan
`− Lω.
`−1 ωTD
`−1 Tω
`1
`1
`
`Therefore, when ω approaches infinity,
`
`(cid:4) G( jω)GPD( jω) < −180
`◦.
`
`(15)
`
`Hence, if TD > T/KKP and KP > 1/K, then by the Nyquist crite-
`rion, the closed-loop system is unstable. This analysis also con-
`firms that some PID mapping formulas, such as the
`Ziegler-Nichol (Z-N) formula obtained from the step-response
`method, in which KP = (1.2(T/L)) (1/K) and TD is proportional
`to L, are valid for only a limited range of values of the T/L ratio.
`As an example, consider plant (10) with K = 10, T = 1 s, and
`L = 0.1 s [7]. Control by means of a PI controller with KP =
`0.644 > 1/K and TI = 1.03 s yields reasonable stability margins
`and time-domain performance, as seen in Figures 2 and 3 (Set 1,
`red curves). However, when a differentiator is added, gradually
`
`(cid:4)
`

`

`Linear Lowpass Filter
`The filtering remedy most commonly adopted is to cascade
`the differentiator with a first-order, lowpass filter, a technique
`often used in preprocessing for data acquisition. Hence, the
`derivative term becomes
`˜GD(s) = KP
`
`TDs
`1 + TD
`β s
`
`,
`
`(16)
`
`where β is a constant factor. Most industrial PID hardware
`provides a value ranging from 1 to 33, with the majority
`falling between 8 and 16 [12]. A second-order Butterworth
`filter is recommended in [13] if further attenuation of high-
`frequency gains is required. Sometimes, the lowpass filter is
`cascaded to the entire GPID in internal-model-control (IMC)-
`based design, which therefore leads to more sluggish tran-
`sients.
`
`Velocity Feedback
`Because a lowpass filter does not completely remove, but
`rather averages, impulse derivative signals caused by sud-
`den changes of the setpoint or disturbance, modifications of
`the unity negative feedback PID structure are of interest [8].
`To block the effect of sudden changes of the setpoint, we
`consider a variant of the standard feedback. This variant uses
`
`Set 1
`PIDeasy
`Set 2
`Set 3
`
`Set 3: TD = 0.2
`
`Set 1:
`Gain Margin: 7.75757
`Phase Margin: 53.37157
`
`PIDeasy:
`Gain Margin: 9.16494
`Phase Margin: 64.72558
`
`Set 2:
`Gain Margin: 3.41323
`Phase Margin: 86.39067
`
`Set 3:
`Gain Margin: −2.26496
`Phase Margin: – ∞
`
`PIDeasy: TD = 0.0303
`
`−2.0
`−1.5
`Phase (deg.)
`
`−1.0
`
`−0.5
`
`0.0
`
`×102
`
`FIGURE 2 Destabilizing effect of the derivative term, measured in the frequency domain by GM and PM. Adding a derivative term increases
`both the GM and PM, although raising the derivative gain further tends to reverse the GM and destabilize the closed-loop system. For
`example, if the derivative gain is increased to 20% of the proportional gain (TD = 0.2 s), the overall open-loop gain becomes greater than
`2.2 dB for all ω. At ω = 30 rad/s, the phase decreases to −π while the gain remains above 2.2 dB. Hence, by the Nyquist criterion, the
`closed-loop system is unstable. It is interesting to note that MATLAB does not compute the frequency response as shown here, since MAT-
`LAB handles the transport delay factor e−j ωL in state space through a Padé approximation.
`
`FEBRUARY 2006 « IEEE CONTROL SYSTEMS MAGAZINE 35
`
`increasing TD from zero improves both GM and PM. The GM
`peaks when TD approaches 0.03 s; this value of TD maximizes
`the speed of the transient response without oscillation. Howev-
`er, if TD is increased further to 0.1 s, the GM deteriorates and
`the transient exhibits oscillation. In fact, the closed-loop system
`can be destabilized if TD increases to 0.2 with T/KKP = 0.155.
`Hence, care needs to be taken to tune and use the derivative
`term properly when the plant is subject to delay.
`This destabilizing phenomenon can contribute to difficul-
`ties in designing PID controllers. These difficulties help explain
`why 80% of PID controllers in use have the derivative part
`switched off or omitted completely [5]. Thus, the functionality
`and potential of a PID controller is not fully exploited, while
`proper use of a derivative term can increase stability and help
`maximize the integral gain for better performance [11].
`
`Remedies for Derivative Action
`Differentiation increases the high-frequency gain, as shown in
`(9) and demonstrated by the four sets of frequency responses
`in Figure 2. A pure differentiator is not proper or causal.
`When a step change of the setpoint or disturbance occurs, dif-
`ferentiation results in a theoretically infinite control signal. To
`prevent this impulse control signal, most PID software pack-
`ages and hardware modules add a filter to the differentiator.
`Filtering is particularly useful in a noisy environment.
`
`PIDeasy(TM) II - Nichols Chart
`
`Nichols Chart
`
`Nichols Chart
`
`Set 2: TD = 0.1
`
`−3.5
`
`−3.0
`
`−2.5
`
`40
`
`30
`
`20
`
`10
`
`0
`
`−10
`
`−20
`
`Gain (dB)
`
`Set 1: TD = 0
`
`

`

`tively using standard techniques of stability and robust-
`ness analysis.
`Structure (17) is referred to as Type B PID (or PI-D) control,
`structure (18) is known as Type C PID (or I-PD) control, and
`structures (1)–(3) constitute Type A PID control. Types B and C
`introduce more structures, and the need for preselection of, or
`switching between, suitable structures can pose a design chal-
`lenge. To meet this need, PID hardware vendors have developed
`artificial intelligence techniques to suppress overshoots [15], [16].
`Nevertheless, the ideal, parallel, series, and modified PID con-
`troller structures can be found in many software packages and
`hardware modules. Techmation’s Applications Manual [12] docu-
`ments the structures employed in many industrial PID con-
`trollers. Since vendors often recommend their own controller
`structures, tuning rules for a specific structure do not necessarily
`perform well with other structures. Readers may refer to [17] and
`[18] for detailed discussions on the use of various PID structures.
`
`Prefilter
`For setpoint tracking applications, an alternative to using a Type
`B or C structure is to cascade the setpoint with a prefilter that
`has critically damped dynamics. When a step change in the set-
`point occurs, continuous output of the prefilter helps achieve
`soft start and bumpless control [8], [19]. However, a prefilter
`does not solve the problem caused by sudden changes in the
`disturbance since it is not embedded in the feedback loop.
`
`the process variable instead of the error signal for the deriva-
`tive action [14], as in
`
`(17)
`
`y(t),
`
`d d
`
`t
`
`t
`
`e(τ ) dτ − KD
`
`0
`
`u(t) = KP e(t) + KI
`
`where y(t) is the process variable, e(t) = r(t) − y(t) is the error
`signal, and r(t) is the setpoint or reference signal. The last term
`of (17) forms velocity feedback and, hence, an extra loop that
`is not directly affected by a sudden change in the setpoint.
`However, sudden changes in disturbance or noise at the plant
`output can cause the differentiator to produce a theoretically
`infinite control signal.
`
`(cid:11)
`
`Setpoint Filter
`To further reduce sensitivity to setpoint changes and avoid
`overshoot, a setpoint filter may be adopted. To calculate the
`proportional action, the setpoint signal is weighted by a factor
`b < 1, as in [8] and [14]
`
`y(t).
`
`(18)
`
`d d
`
`t
`
`t
`
`0 e(τ ) dτ − KD
`
`(cid:8)
`
`u(t) = KP
`
`(cid:9) + KI
`
`b r(t) − y(t)
`
`(cid:12)
`
`This modification results in a bumpless control signal and
`improved transients if the value of b is carefully chosen [2].
`However, modification (18) is difficult to analyze quantita-
`
`PIDeasy(TM) II - Step and Control Signal Response Dialog
`
`View Both Responses
`
`Sampling Rate: 0.00014 sec.
`
`Set 1
`PIDeasy
`Set 2
`Set 3
`
`Set 1:
`
`Kp: 0.6439
`Ti: 1.0278
`Td: 0
`ITAE: 228.93
`
`PIDeasy:
`
`Kp: 0.6439
`Ti: 1.02781
`Td: 0.03025
`ITAE: 104.86
`
`Set 2:
`
`Kp: 0.6439
`Ti: 1.0278
`Td: 0.1
`ITAE: 247.69
`
`Set 3:
`
`Kp: 0.6439
`Ti: 1.0278
`Td: 0.2
`ITAE: 52,547.83
`
`Step Response Plot
`
`Set 1: TD = 0
`
`Set 2: TD = 0.1
`
`Step Response
`
`1.4
`
`1.2
`
`1.0
`
`0.8
`
`0.6
`
`0.4
`
`0.2
`
`0.0
`
`Output
`
`0.0
`
`0.2
`
`0.4
`
`PIDeasy: TD = 0.0303
`
`0.6
`0.8
`Time (Sec.)
`
`1.0
`
`1.2
`
`1.4
`
`Set 3: TD = 0.2
`
`FIGURE 3 Destabilizing effect of the derivative term, confirmed in the time domain by the closed-loop step response. Although increasing the
`derivative gain initially decreases the oscillation, this trend soon reverses and the oscillation grows into instability.
`
`36 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2006
`
`

`

`Nonlinear Median Filter
`Another method for smoothing the derivative action is to use
`a median filter [7], which is nonlinear and widely applied in
`image processing. Such a filter compares several data points
`around the current point and selects their median for the con-
`trol action. Consequently, unusual or unwanted spikes result-
`ing from a step command, noise, or disturbance are removed
`completely. Median filters are easily realized, as illustrated in
`Figure 4, since almost all PID controllers are now implemented
`in a digital processor. Another benefit of this method is that
`extra parameters are not needed to devise the filter. A median
`filter outperforms a prefilter as the median filter is embedded
`in the feedback loop and, hence, can deal with sudden
`changes in both the setpoint and the disturbance; a median fil-
`ter may, however, overly smooth underdamped processes.
`
`Design Objectives and Methods
`
`Design Objectives and Existing Methods
`Excellent summaries of PID design and tuning methods can be
`found in [4], [8], [20], and [21]. While matters concerning com-
`missioning and maintenance (such as pre- and postprocessing as
`well as fault tolerance) also need to be considered in a complete
`PID design, controller parameters are usually tuned so that the
`closed-loop system meets the following five objectives:
`1) stability and stability robustness, usually measured in the
`frequency domain
`2) transient response, including rise time, overshoot, and set-
`tling time
`3) steady-state accuracy
`4) disturbance attenuation and robustness against environ-
`mental uncertainty, often at steady state
`5) robustness against plant modeling uncertainty, usually
`measured in the frequency domain.
`Most methods target one objective or a weighted compos-
`ite of the objectives listed above. With a given objective,
`design methods can be grouped according to their underlying
`nature listed below [7], [8].
`
`Heuristic Methods
`Heuristic methods evolve from empirical tuning (such as the Z-
`N tuning rule), often with a tradeoff among design objectives.
`Heuristic search now involves expert systems, fuzzy logic,
`neural networks, and evolutionary computation [19], [22].
`
`Frequency Response Methods
`Frequency-domain constraints, such as GM, PM, and sensi-
`tivities, are used to synthesize PID controllers offline [2],
`[3]. For real-time applications, frequency-domain measure-
`ments require time-frequency, localization-based methods
`such as wavelets.
`
`Analytical Methods
`Because of the simplicity of PID control, parameters can be
`derived analytically using algebraic relations between a plant
`
`PID
`
`PD
`
`P
`
`PI
`
`PB
`
`PB
`
`TI
`
`PB
`
`TI
`
`TD
`
`PB
`
`TD
`
`model and a targeted closed-loop transfer function with an
`indirect performance objective, such as pole placement, IMC,
`or lambda tuning. To derive a rational, closed-loop transfer
`function, this method requires that transport delays be
`replaced by Padé approximations.
`
`derivative = (error -previous_error) / sampling_period;
`if (derivative > max_d)
`new_derivative = max_d;
`else if (derivative < min_d)
`new_derivative = min_d;
`
`// median found
`
`// median found
`
`else
`
`new_derivative=derivative;
`
`// median found
`
`// for next cycle
`
`// for next cycle
`
`if (derivative > previous_derivative) {
`max_d=derivative;
`min_d=previous_derivative;
`} else {
`max_d=previous_derivative;
`min_d=derivative;
`
`} p
`
`revious_derivative = derivative;
`
`FIGURE 4 Pseudocode for a three-point median filter to illustrate the
`mechanism of complete removal of impulse spikes. Median filters
`are widely adopted in image processing but not yet in control sys-
`tem design. This nonlinear filter completely removes extraordinary
`derivative values resulting from sudden changes in the error signal,
`unlike a lowpass filter, which averages past values.
`
`TABLE 2 ABB’s Easy-Tune PID formulas mapping the three
`parameters K, T, and L of the first-order delayed plant (10)
`to coefficients of P, PI, PID, and PD controllers, respectively
`[23]. The formulas are obtained by minimizing the
`integral of time-weighted error index, except the PD formula
`for which empirical estimates are used. Usually expressed
`in percentage, PB = (Umax − Umin)/K P is the proportional
`band, where Umax and Umin are the upper and lower
`saturation levels of the control signal, respectively, and
`|Umax − Umin| is usually normalized to one.
`(cid:2)
`Mode
`Action
`Value
`(cid:2)
`(cid:2)
`
`(cid:3)1.084
`(cid:3)0.977
`(cid:3)0.68
`(cid:2)
`(cid:3)0.947
`(cid:3)0.738
`(cid:3)0.995
`(cid:3)0.947
`(cid:2)
`(cid:3)0.995
`
`L T
`
`L T
`
`L T
`
`2.04K
`
`1.164K
`
`T
`40.44
`
`L T
`
`L T
`
`0.7369K
`
`(cid:2)
`(cid:2)
`
`(cid:2)
`
`T
`51.02
`T
`157.5
`
`T
`157.5
`
`0.5438K
`
`L T
`
`L T
`
`L T
`
`FEBRUARY 2006 « IEEE CONTROL SYSTEMS MAGAZINE 37
`
`

`

`Numerical Optimization Methods
`Optimization-based methods can be regarded as a special type
`of optimal control. PID parameters are obtained by numerical
`optimization for a weighted objective in the time domain.
`Alternatively, a self-learning evolutionary algorithm (EA) can
`be used to search for both the parameters and their associated
`structure or to meet multiple design objectives in both the
`
`10−2
`
`10−1
`
`100
`
`101
`
`102
`
`103
`
`10−2
`
`10−1
`
`100
`L/T
`
`101
`
`102
`
`103
`
`15
`
`10
`
`5
`10−3
`80
`
`70
`
`60
`
`50
`10−3
`
`Gain Margin (dB)
`
`Phase Margin (°)
`
`FIGURE 5 Gain and phase margins resulting from PIDeasy designs for first-
`order delayed plants with various L/T ratios. While requirements of fast
`transient response, no overshoot, and zero steady-state error are accom-
`modated by time-domain criteria, multiobjective design goals provide
`◦
`frequency-domain margins in the range of 9–11 dB and 65–66
`.
`
`time and frequency domains [19], [22].
`Some design methods can be computerized, so that designs
`are automatically performed online once the plant is identified;
`hence, these designs are suitable for adaptive tuning. While
`PID design has progressed from analysis-based methods to
`numerical optimization-based methods, there are few tech-
`niques that are as widely applicable as Z-N tuning [2], [3]. The
`most widely adopted initial tuning methods are based
`on the Z-N empirical formulas and their extensions, such
`as those shown in Table 2 [23]. These formulas offer a
`direct mapping from plant parameters to controller coef-
`ficients.
`Over the past half century, researchers have sought the
`next key technology for PID tuning and modular realiza-
`tion [6]. With simulation packages widely available and
`heavily adopted, computerizing simulation-based designs
`is gaining momentum, enabling simulations to be carried
`out automatically so as to search for the best possible PID
`settings for the application at hand [22]. By using a com-
`puterized approach, multiple design methods can be com-
`bined within a single software or firmware package to
`support various plant types and PID structures.
`
`A Computerized Simulation Approach
`PIDeasy [7] is a software package that uses automatic sim-
`ulations to search globally for controllers that meet all five
`design objectives in both the time and frequency domains.
`The search is initially performed offline in a batch mode
`[19] using artificial evolution techniques that evolve both
`
`TABLE 3 Multioptimal PID settings for normalized typical high-order plants. Since PIDeasy’s search priorities are time-domain
`tracking and regulation, the corresponding gain and phase margins are given to assess frequency-domain properties.
`
`Plants
`
`G1(s) =
`
`1
`(s + 1)α
`
`G2(s) =
`
`1
`(s + 1)(1 + αs)(1 + α2s)(1 + α3s)
`
`G3(s) = 1 − αs
`(s + 1)3
`
`G4(s) =
`
`1
`(1 + sα)2 e
`
`−s
`
`PID Coefficients
`Ti (s)
`
`Td (s)
`
`1.0
`1.61
`2.13
`2.61
`4.31
`
`1.03
`1.08
`1.36
`
`2.15
`2.18
`2.23
`2.30
`2.39
`2.58
`
`0.43
`0.59
`1.07
`3.49
`8.32
`16.35
`
`0.0022
`0.14
`0.28
`0.43
`1.01
`
`0.04
`0.07
`0.17
`
`0.31
`0.33
`0.39
`0.47
`0.57
`0.72
`
`0.12
`0.17
`0.26
`0.49
`0.92
`1.59
`
`Kp
`
`92.1
`1.95
`1.12
`0.83
`0.50
`
`5.53
`2.87
`1.19
`
`1.03
`0.96
`0.79
`0.63
`0.48
`0.36
`
`0.23
`0.30
`0.49
`1.04
`1.42
`1.65
`
`Resultant Margins
`◦
`GM (dB)
`PM (
`)
`∞
`∞
`26.8
`13.9
`9.05
`
`102
`62.4
`60.7
`61
`58.9
`
`52.8
`38.6
`19.1
`
`19.4
`16.6
`13
`7.52
`7.45
`2.69
`
`10.4
`10.4
`10.5
`15
`24.2
`32.8
`
`68.7
`66.3
`62.6
`
`61.2
`61.6
`62.4
`50.9
`58.6
`40.4
`
`66
`65.8
`65.6
`62.4
`62.1
`62.1
`
`α = 1
`α = 2
`α = 3
`α = 4
`α = 8
`α = 0.1
`α = 0.2
`α = 0.5
`α = 0.1
`α = 0.2
`α = 0.5
`α = 1.0
`α = 2.0
`α = 5.0
`α = 0.1
`α = 0.2
`α = 0.5
`α = 2.0
`α = 5.0
`α = 10
`
`38 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2006
`
`

`

`ear projection linking the starting and ending points of the
`operating envelope, as illustrated by node 2 in Figure 6 [22].
`Similarly, two more controllers can be added at nodes or
`setpoints 1 and 3, forming a pseudolinear controller net-
`work comprised of three PIDs to be interweighted by sched-
`uling functions S1(y), S2(y), and S3(y), examples of which are
`shown in Figure 7.
`
`0.74
`
`3
`
`0.49
`
`2
`
`Δdmax
`
`0.31
`
`1
`
`1
`0.9
`0.8
`
`0.7
`0.6
`
`0.5
`0.4
`
`0.3
`0.2
`
`0.1
`
`Output (mol/l )
`
`0
`
`0
`
`1
`
`2
`
`3
`Input (l/h)
`
`4
`
`5
`
`6
`
`controller parameters and their associated structures. For practi-
`cal simplicity and reliability, the standard PID structure is main-
`tained as much as possible, while allowing augmentation with
`either lowpass or median filtering for the differentiator and with
`explicit antiwindup for the integrator. The resulting designs are
`then embedded in the PIDeasy package. Further specific tuning
`can be continued by local, fast numerical optimization if the
`plant differs from its model or data used in the initial design.
`
`First-Order Delayed Plants
`An example of PIDeasy for a first-order delayed plant is
`shown in Figures 2 and 3. To assess the robustness of design
`using PIDeasy, GMs and PMs resulting from designs for
`plants with various L/T ratios are shown in Figure 5 [19].
`While requirements of fast transient response, no overshoot,
`and zero steady-state error are accommodated by time-
`domain criteria, PIDeasy’s multiobjective goals provide fre-
`◦
`.
`quency-domain margins in the range of 9–11 dB and 65–66
`
`Higher Order Plants
`For higher order plants, we obtain multioptimal designs for
`the 20 benchmark plants [24]
`
`,
`
`,
`
`G1(s) = 1
`α = 1, 2, 3, 4, 8,
`(s + 1)α
`G2(s) =
`1
`(s + 1)(1 + αs)(1 + α2s)(1 + α3s)
`α = 0.1, 0.2, 0.5,
`G3(s) = 1 − αs
`α = 0.1, 0.2, 0.5, 1, 2, 5,
`(s + 1)3
`G4(s) =
`α = 0.1, 0.2, 0.5, 2, 5, 10.
`1
`−s,
`(1 + sα)2 e
`
`,
`
`(19)
`
`(20)
`
`(21)
`
`(22)
`
`FIGURE 6 Operating trajectory (bold curve) of the nonlinear chemical
`process (23) for setpoints ranging from 0 to 1 mol/(cid:8), as given by
`(24). A PID controller is first placed at y = 0.49 (node 2) by using
`the maximum distance from the nonlinear trajectory to the linear
`projection (thin dotted line) linking the starting and ending points of
`the operating envelope. Similarly, two more controllers can be
`added at nodes 1 and 3, forming a pseudo-linear controller network
`comprised of three PIDs. Without the need for linearization, these
`PID controllers can be obtained individually by PIDeasy or other PID
`software directly through step-response data, or obtained jointly by
`using an evolutionary algorithm [22].
`
`The resulting designs and their corresponding gain and PMs
`are summarized in Table 3.
`
`S1
`
`S2
`
`S3
`
`0.31
`
`0.49
`
`0.74
`
`1
`
`0.8
`
`0.6
`
`0.4
`
`0.2
`
`0
`
`Weighting
`
`0
`
`0.1
`
`0.2
`0.3
`0.4
`0.5
`0.6
`0.7
`Scheduling Variable y (mol/l)
`
`0.8
`
`0.9
`
`FIGURE 7 Fuzzy logic membership-like scheduling functions S1(y ),
`S2(y ), and S3(y ) for individual PID controllers contributing to the
`PID network at nodes 1, 2, and 3, respectively. Due to nonlinearity,
`these functions are often asymmetric. Similar to gain scheduling, lin-
`ear interpolation suffices for setpoint scheduling.
`
`FEBRUARY 2006 « IEEE CONTROL SYSTEMS MAGAZINE 39
`
`Setpoint-Scheduled PID Network
`(cid:14)
`Consider the constant-temperature reaction process
`= −Ky2(t) + 1
`d − y(t)
`V
`
`u(t),
`
`(23)
`
`(cid:13)
`
`dy(t)
`dt
`
`where
`y(t) = concentration in the outlet stream (mol/(cid:8)),
`u(t) = flow rate of the feed stream ((cid:8)/h),
`K = rate of reaction ((cid:8)/mol-h),
`V = reactor volume ((cid:8) ),
`d = concentration in the inlet stream (mol/(cid:8) ).
`The setpoint, equilibrium, or steady-state operating trajectory
`of the plant is governed by
`u y − d
`Ky2 + 1
`V
`V
`
`u = 0.
`
`(24)
`
`For setpoints ranging from 0 to 1 mol/(cid:8), an initial PID
`controller can be placed effectively at y = 0.49 by using the
`maximum distance from the nonlinear trajectory to the lin-
`
`

`

`The PID controller centered at node 2 can be obtained by
`PIDeasy or other PID software directly through step-response
`data without the need for linearization at the current operating
`
`0.8
`
`0.6 0.53
`0.4
`
`0.2
`
`0
`
`0
`
`0.5
`
`1
`
`1.5
`
`2
`
`2.5
`Time (h)
`
`3
`
`3.5
`
`4
`
`4.5
`
`5
`
`0
`
`0.5
`
`1
`
`1.5
`
`2
`
`2.5
`Time (h)
`
`3
`
`3.5
`
`4
`
`4.5
`
`5
`
`0246
`
`Output (mol l−1)
`
`Control Signal (l h−1)
`
`FIGURE 8 Performance of the pseudolinear PID network applied to
`the nonlinear process example (23). To validate tracking perfor-
`mance using a setpoint that is not originally used in the design
`process, the setpoint r = 0.53 mol/(cid:8) is used to test the control sys-
`tem. The controller network tracks this setpoint change accurately
`without oscillation and rejects a 10% load disturbance occurring
`during [3, 3.5] h.
`
`where p denotes the derivative operator.
`To validate tracking performance using a setpoint that is
`not originally used in the design process, the setpoint r = 0.53
`mol/(cid:8) is used to test the control system. The response is
`shown in Figure 8, where a 10% disturbance occurs during [3,
`3.5] h, confirming load disturbance rejection at steady state.
`Figure 9 shows the performance of the network at multiple
`operating levels not originally encountered in the design. If a
`more sophisticated PID network is desirable, the number of
`nodes, controller parameters for each node, and scheduling
`functions can be optimized globally in a single design process
`by using an EA [22].
`It is known that gain scheduling provides advantages over
`continuous adaptation in most situations [8]. The setpoint-
`scheduled network utilizes these advantages of gain schedul-
`ing. Furthermore, by bumpless scheduling, the network does
`not require discontinuous switching between various con-
`troller structures.
`
`Discussion and Conclusions
`PID is a generally applicable control technique that
`derives its success from simple and easy-to-under-
`stand operation. However, because of limited infor-
`mation exchange and problem analysis, there
`remain misunderstandings between academia and
`industry concerning PID control. For example, the
`message that increasing the derivative gain leads to
`improved transient response and stability is often
`wrongly expounded. These misconceptions may
`explain why the argument exists that academically
`proposed PID tuning rules sometimes do not work
`well on industrial controllers. In practice, therefore,
`switching between different structures and func-
`tional modes is used to optimize transient response
`and meet multiple objectives.
`Difficulties in setting optimal derivative action
`can be eased by complete understanding and careful
`tuning of the D term. Median filtering, which is
`widely adopted for preprocessing in image process-
`ing but yet to be adopted in controller design, is a
`convenient tool for solving problems that the PI-D
`and I-PD structures are intended to address. A medi-
`an filter outperforms a lowpass filter in removing
`impulse spikes of derivative action resultin