Novel Multiscale Control Scheme for Nonminimum-Phase Processes

Article pubs.acs.org/IECR

Novel Multiscale Control Scheme for Nonminimum-Phase Processes Jobrun Nandong*,† and Zhuquan Zang‡ †

Department of Chemical Engineering and ‡Department of Electrical and Computer Engineering, Curtin University Sarawak, 98009 Miri, Malaysia ABSTRACT: Nonminimum-phase (NMP) processes are frequently encountered in chemical plants, where the presence of right half-plane zeros and time delays often imposes limitations on the achievable control performance. In this article, a new multiscale control scheme for controlling NMP processes is introduced. The novelty of the proposed scheme is to decompose a given plant into a sum of basic factors/modes having distinct speeds of responses. Then, an individual subcontroller is specifically designed to control each of these plant modes. Finally, an overall multiscale controller is synthesized by combining all of the subcontrollers in a way that enhances good cooperation among the different modes, which is an essential feature for good control performance and robustness. Extensive numerical studies demonstrate that the proposed scheme can provide better nominal performance and performance robustness than some well-established control schemes.

1. INTRODUCTION Nonminimum-phase (NMP) processes, or processes with right half-plane (RHP) zeros (including those with time delays), are frequently encountered in chemical plants. Typically, an NMP process exhibits an initial response to an input that is in the opposite direction to the final steady-state response, often called inverse-response behavior. Inverse-response systems are normally difficult to control because of the RHP zeros, which often impose limitations on the achievable closed-loop performance. The associated closed-loop performance usually degenerates as the RHP zeros get slower.1 Moreover, the restriction on the control performance becomes more severe when the systems have RHP poles as well as RHP zeros.2 In addition to the speeds (locations) of RHP zeros, it has been reported that the number of RHP zeros also plays an important role in determining achievable control performance.3 Generally, the effect of RHP zeros on the stability of control systems is well-known; a specific example is adaptive control based on the minimum-variance law, for which it has been known that the RHP zeros can cause instability.4 Over the years, a variety of control approaches have been proposed to improve the control performance of inverse-response systems. One common approach is based on the application of standard single-loop feedback control, which normally uses the traditional proportional−integral−derivative (PID) controller. In this standard approach, researchers often aim to improve the control performance through applications of specific tuning procedures, such as PID controller tuning procedures.5−8 Some of the controller tuning procedures are based on nonlinear optimization techniques, such as particle swarm PI/PD controller tuning9 and parameter optimization tuning.10 For the standard single-loop control scheme, the main benefit of applying a PID controller comes from the derivative mode, which can reduce the negative effect of inverse-response dynamics.11 Unfortunately, this advantage is normally gained at the cost of making the controller less robust to plant/model uncertainties. To increase the robustness of a PID controller, it is often necessary to detune the controller at the nominal plant conditions. This, in turn, often results in sluggish performance of the PID controller when applied to real plants. © XXXX American Chemical Society

Another approach to controlling inverse-response systems is based on the well-known Smith predictor, introduced by Smith12 in the late 1950s to control systems characterized by long time delays. The principle of the Smith predictor scheme is to eliminate the time-delay component from the closed-loop characteristic equation, which, in turn, enables the controller to be designed as if the plant were delay-free. For inverse-response systems, various modified Smith predictor schemes have been developed over the years. Most of these modified Smith predictor schemes share one common principle, which is to eliminate the inverse-response dynamics from the feedback loop. This elimination, in turn, allows the controller to be designed based on the minimum-phase dynamic only. Some examples of these modified Smith predictor schemes are inverse-response compensator (IRC) schemes13,14 for stable inverse-response processes, IRC schemes with a nonminimum-phase predictor15 derived on the basis of the partial internal model principle16 for unstable inverse-response systems, and the modified Smith predictor17 for integrating inverse-response systems. Another control scheme that can also be used to control stable inverse-response systems is the internal model control (IMC) scheme.18 The IMC and Smith predictor schemes are similar in principle, in that they aim to eliminate the undesirable dynamics (inverse-response/time-delay component) from the closed-loop characteristic equation. In fact, through a suitable transformation, the IMC scheme can be shown to be interchangeable with the original Smith predictor scheme.19 Similarly to the original Smith predictor, IMC is also not directly applicable to unstable processes because of internal instability.20 In this article, a novel multiscale control (MSC) scheme for inverse-response (NMP) systems is introduced. The principle of the proposed MSC scheme is to decompose a given plant into a sum of basic modes or factors with distinct speeds of responses, that is, different time scales. Following plant decomposition, Received: October 16, 2012 Revised: April 17, 2013 Accepted: May 20, 2013

A

dx.doi.org/10.1021/ie302839v | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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an individual subcontroller is specifically designed to control each plant mode. Then, an overall controller is synthesized through a cascade-based combination of all of the subcontrollers; the subcontrollers can be tuned in such a way that the sum of the integral square errors (ISEs) associated with the subcontrollers is minimized. It is important to note that all of the subcontrollers must be combined in a manner that they can function cooperatively, a combination based on the cascade control structure. From the proposed multiscale control perspective, achieving good cooperation among the different plant modes is vital to improving the control performance and performance robustness. In contrast to the proposed MSC design principle, the PID, LQG, and IMC controllers used in conventional single-loop control schemes are normally designed based on the dominant (slowest) plant modes, and the fast modes are simply ignored. Therefore, these conventional control schemes can be interpreted as using only a single controller to control all of the different plant modes. It is worth noting that the proposed MSC scheme is applicable not only to stable minimum-phase processes but also to integrating/unstable NMP processes (including those with time delays). The rest of this article is structured as follows: In section 2, some preliminary details are provided. The block diagrams and design procedure for the proposed MSC scheme are described in section 3. In section 4, the effectiveness of the proposed MSC scheme is illustrated using a number of single-input−singleoutput (SISO) examples. Finally, conclusions and suggestions for future work are highlighted in section 5.

Figure 1. Feedback control block diagram: (a) standard single-loop structure and (b) internal model control (IMC) structure.18

set of plant modes or factors. To design the IMC controller for a NMP system, the nominal process model P̅ must be factorized first into the minimum-phase factor M−1 and the nonminimumphase factor N

2. PRELIMINARIES 2.1. Nonminimum-Phase Systems. A nonminimum-phase (NMP) system can be represented by two or more conflicting factors having different speeds of response to an input. For example, consider an inverse-response system with two conflicting factors expressed as P(s) =

k1 k2 + τ1s + 1 τ2s + 1

P ̅(s) = N (s) M −1(s)

For a step input, a robust IMC controller is given by Q (s ) = M (s ) F (s )

(1)

There are two tuning parameters for the IMC structure, namely, λ and n, where λ is chosen for robustness (larger value is more robust but leads to slower dynamic response) and n is chosen to make the controller proper. The IMC structure can be converted into the standard single-loop feedback control scheme shown in Figure 1a. In terms of the standard single-loop feedback control, the corresponding IMC controller (K) is given by

(2)

Note that the two conflicting factors in eq 1 are actually acting at two different time scales (i.e., with distinct speeds of response) to the same input change. For this reason, a given system P can be decomposed by partial fraction expansion into a sum of several factors or modes with distinct speeds of responses as follows P(s) = P0(s) + P1(s) + ··· + Pn(s)

(5)

where F is a low-pass filter. For the case of a step input, the following structure14 is obtained 1 F (s ) = , λ>0 (λs + 1)n (6)

where |k1| > |k2| such that k1/k2 < 0 and τ1 > τ2, which means the dominant factor with gain k1 has slower dynamics than the nondominant factor with gain k2. For eq 1 to exhibit an inverseresponse behavior, the following condition is applied (k1τ2 + k 2τ1) 0], η̅(Pj) is the number of poles Pj in the closed right half-plane [Re(s) ≥ 0] counted according to multiplicity, and wno denotes the winding number evaluated on the standard Nyquist contour indented into the right half-plane around any imaginary axis poles of P1 and P2. The notation X* denotes the conjugate system XT(−s) for real rational systems, and ∥X∥∞ denotes the L∞ norm of the transfer matrix X, given by ∥X∥∞ = supω∈R{σmax[X(jω)]}, where σmax(X) denotes the maximum singular value of a matrix X. By using the Vinnicombe distance, one can quantify the extent of plant/model mismatch (or modeling error). For example, the modeling error between a nominal plant (P̅ ) and the corresponding perturbed plant (P) is given by δv(P̅,P).

3. NOVELTY OF THE PROPOSED MULTISCALE CONTROL SCHEME Figure 3 shows the block diagrams of two-layer multiscale control (MSC) schemes: a direct scheme (Figure 3a) and an indirect scheme (Figure 3b). For the two-layer MSC scheme, it is assumed that a given plant P can be decomposed into a sum of two factors with distinct time scales: P = P0 + P1, where P0 is slower than P1. Note that P0 is called the outermost factor corresponding to the outermost control loop, and P1 denotes the inner-layer factor corresponding to the inner control loop. In general, for a plant that can be decomposed into a sum of n + 1 modes, there are n inner-layer factors and one outermost factor. Referring to Figure 3a,b, K0 and K1 denote the subcontrollers with distinct time scales or speed of responses; Gd denotes the output disturbance transfer function; Di, Do, R, and Y denote the signals corresponding to the input disturbance, output disturbance, set point, and controlled output, respectively; and C



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Figure 4. Multiscale control scheme: (a) three-loop, (b) reduced two-loop, and (c) equivalent single-loop block diagrams.

block diagram as depicted in Figure 4b. The same procedure can be repeated by simplifying the remaining inner loop corresponding to subcontroller K1. This final simplification reduces the three-layer MSC block diagram to an equivalent standard singleloop feedback control scheme as illustrated in Figure 4c. Based on Figure 4c, a transfer function called the augmented overall plant transfer function is expressed as

of control loopsthe process is viewed as a collection of some basic modes. Ideally, the basic principle of the proposed MSC scheme is to utilize all dynamics information (represented as factors or modes) about a given plant P with the aim of enhancing cooperation among these different plant modes. To achieve good cooperation among the different plant modes, it is important to assemble the subcontrollers based on the cascade structure. Additionally, each subcontroller can be interpreted as being designed to control a specific plant mode, where the slower mode commands the faster mode. By enhancing good cooperation among the different plant modes, a substantial performance (including robustness) improvement can be made over that of the standard single-loop feedback control scheme. For the standard single-loop feedback control scheme where only a single controller is used, good cooperation among the different plant modes might not be achievable, and subsequently, this could result in rather poor closed-loop performance/robustness. 3.1. Block Diagram Reduction of MultiScale Control Scheme. Consider Figure 4, which illustrates a three-layer direct MSC scheme. In this case, there are three subcontrollers that must be designed to operate at different time scales; controller K2 must be within a faster control loop than controller K1, which, in turn, must be designed to be faster than controller K0. As illustrated in Figure 4, after simplifying the innermost layer corresponding to the fastest subcontroller K2, the three-layer MSC block diagram can be reduced to an equivalent two-layer

Pc(s) = G1(s) G2(s) P(s)

(13)

where the inner-layer closed-loop transfer functions in eq 13 can be expressed as G1(s) =

K1(s) U (s ) = 1 + G2(s) K1(s) W1(s) C0(s)

(14)

G2(s) =

K 2(s) U (s ) = 1 + K 2(s) W2(s) C1(s)

(15)

Meanwhile, for the direct MSC scheme, the multiscale predictors are given by ⎡ W1(s) ⎤ ⎡ P1̅(s) ⎤ ⎥ ⎢ ⎥=⎢ ⎢⎣W2(s)⎥⎦ ⎢⎣ P2̅ (s)⎥⎦

(16)

The overall multiscale controller, Kmsc, can be expressed as K msc(s) = K 0(s) G1(s) G2(s)

(17)

3.2. Generalized Multiscale Control Scheme. For the generalized MSC scheme, consider a plant P that can be D



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Figure 5. Multilayer [(n + 1)-layer] multiscale control scheme: (a) (n + 1)-loop and (b) equivalent single-loop block diagrams.

decomposed into a sum of n + 1 factors, namely, P = P0 + P1 + ··· + Pn. The generalized MSC block diagram is depicted in Figure 5a. This multilayer MSC block diagram can be reduced to an equivalent standard single-loop feedback control as shown in Figure 5b. Based on Figure 5, a vector of inner-layer closed-loop transfer functions corresponding to a number n of subcontrollers in the inner loops, G ∈ Rn, is expressed as

Next, a vector of augmented inner-layer transfer functions H ∈ Rn−1 can be written as ⎡ n ⎤ ⎢ ∏ [Gj(s)]W1(s) ⎥ ⎢ j=2 ⎥ ⎢ ⎥ ⎡ H1(s) ⎤ ⎢ n ⎥ ⎢ ⎥ ⎢ ∏ [Gj(s)]W2(s) ⎥ ⎢ H2(s) ⎥ ⎢ j = 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⋮ H=⎢ ⋮ ⎥=⎢ ⎥, ⎢ ⎥ ⎢ n ⎥ ⎢ Hn − 2(s)⎥ ⎢ ∏ [G (s)]W (s)⎥ j n−2 ⎢ H (s ) ⎥ ⎢ ⎥ j=n−1 ⎣ n−1 ⎦ ⎢ ⎥ ⎢ n ⎥ ⎢ ∏ [Gj(s)]Wn − 1(s) ⎥ ⎢⎣ j = n ⎥⎦

⎤ ⎡ K1(s) ⎥ ⎢ ⎢ 1 + K1(s) W1(s) G2(s) G3(s)··· Gn − 1(s) Gn(s) ⎥ ⎥ ⎡ G1(s) ⎤ ⎢ K 2(s) ⎥ ⎥ ⎢ ⎢ ⎢ G2(s) ⎥ ⎢ 1 + K 2(s) W2(s) G3(s) G4(s)··· Gn − 1(s) Gn(s) ⎥ ⎥ ⎥ ⎢ ⎢ ⎥, G=⎢ ⋮ ⎥=⎢ ⋮ ⎥ ⎥ ⎢ ⎢ G s ( ) K s ( ) ⎥ n−1 ⎢ n−1 ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ 1 + K n − 1(s) Wn − 1(s) Gn(s) ⎣ Gn(s ⎦ ⎢ ⎥ ⎥ ⎢ K n(s) ⎥ ⎢ 1 + K n(s) Wn(s) ⎦ ⎣

(20) (18)

n>1

Significantly, the transfer function H is useful for designing the inner-layer subcontroller Ki, ∀i ∈ {1, 2, 3, ..., n}. For the generalized MSC scheme, the augmented overall plant transfer function can be written as

Alternatively, eq 18 can be rewritten in a compact form as ⎡ ⎤ K1(s) ⎢ ⎥ n ⎢ 1 + K1(s) W1(s)[∏ j = 2 Gj(s)] ⎥ ⎢ ⎥ K 2(s) ⎢ ⎥ ⎢ 1 + K (s) W (s)[∏n G (s)] ⎥ 2 2 j=3 j ⎢ ⎥ ⎢ ⎥, G= ⋮ ⎢ ⎥ ⎢ ⎥ K n − 1(s) ⎢ ⎥ n ⎢ 1 + K n − 1(s) Wn − 1(s)[∏ j = n Gj(s)] ⎥ ⎢ ⎥ K n(s) ⎢ ⎥ ⎢ ⎥ 1 + K n(s) Wn(s) ⎣ ⎦

n>1

n

Pc(s) =

∏ [Gj(s)]P(s) (21)

j=1

Alternatively, the augmented overall plant transfer function in eq 21 can be expressed in terms of the inner-layer subcontrollers K1, K2, ..., Kn, leading to

n>1

n

Pc(s) =

{∏i = 1 [K i(s)]}P(s) n

n

1 + ∑i = 1 {∏ j = i [Kj(s)]Wi (s)}

(22)

For the case of direct MSC scheme, the multiscale predictors are chosen as

(19) E


Industrial & Engineering Chemistry Research ⎡ W (s ) ⎤ ⎡ P ̅ (s ) ⎤ ⎢ 1 ⎥ ⎢1 ⎥ ⎢W2(s)⎥ ⎢ P2̅ (s)⎥ W=⎢ ⎥=⎢ ⎥ ⎢⋮ ⎥ ⎢⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣Wn(s)⎦ ⎣ Pn̅ (s)⎦

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the proposed MSC scheme to a typical real integrating process with delay for which, upon approximating the delay using the firstorder Padé formula, the dynamics difference is less than 1 order of magnitude. Example 6 is used to demonstrate the applicability/ effectiveness of the proposed MSC scheme to a high-order process where the controller is designed based on a reduced-order model. The nominal performance of the proposed MSC scheme and its performance robustness are compared with those of some well-established control schemes: (1) internal model control (IMC), (2) standard single-loop feedback control using a linear quadratic Gaussian (LQG) controller, (3) inverse-response compensation (IRC), and (4) standard single-loop feedback control using a classical PID controller. Note that all of the controllers (IMC, LQG, and P/PI/PID) used in the examples were designed using the SISO Design Tool available in Matlab Control System Toolbox. The dynamic simulations were performed using the Matlab Simulink, where the stiff ode15s solver was used due to the multiscale dynamics of the systems involved. 4.1. Example 1: Stable Inverse-Response System. Assume that a given inverse-response system P can be decomposed into two factors

(23)

where P̅i denotes the nominal model corresponding to plant factor Pi. The overall multiscale controller is given by n

K msc(s) = K 0(s) ∏ Gi(s) i=1

(24)

3.3. Multiscale Control Scheme: Design Procedure. To synthesize the overall multiscale controller, the innermost subcontroller first needs to be designed. The following general procedure can be applied in the controller synthesis: Step 1. Decompose a given nominal plant P̅ into a sum of factors by using partial fraction expansion technique, that is, P̅ = P̅0 + P̅1 + ··· + P̅n. Step 2. Design the innermost subcontroller Kn based on Wn=P̅n where P̅n denotes the nominal model corresponding to the factor Pn. Step 3. Derive the transfer function Gn as in eq 18 or 19, and subsequently derive Hn−1 as in eq 20. The subcontroller Kn−1 is designed based on Hn−1. Repeat this step to synthesize the remaining subcontrollers Kn−3, Kn−4, ..., K1. Step 4. Once all of the subcontrollers in the inner loops have been synthesized, derive the augmented overall plant transfer function Pc as in eq 21 or 22. The outermost subcontroller K0 is then designed based on Pc, where the overall multiscale controller is obtained by 24. Remark. For subcontroller tuning, the following extra steps can be employed: Step 5. First, the subcontrollers are tuned based on the minimum integral of absolute error (IAE) criterion using the Matlab SISO Design Tool. Step 6. Next, controller tuning values obtained in step 5 are refined through a trial-and-error approach to achieve the desired performance and robustness.

P(s) = P0(s) + P1(s)

(25)

where the slow factor (P0) and fast factor (P1) are given by ⎤ ⎡ 1 + a0 ⎡ P0(s)⎤ ⎢ 100(1 + b0)s + 1 ⎥ ⎥ ⎢ ⎥=⎢ ⎢⎣ P1(s) ⎥⎦ ⎢ −0.4(1 + a1) ⎥ ⎥ ⎢ ⎢⎣ (1 + b1)s + 1 ⎥⎦

(26)

In eq 25, ai and bi denote the plant/model mismatch or modeling errors for the gain and time constant, respectively. The nominal model P̅ corresponding to ai = bi = 0, i = 1, 2, is simply given by the sum of its nominal factors, namely, P̅(s) = P̅0 + P̅1. Notice that the RHP zero of the nominal plant is located at s = 0.0154, indicating that the system is indeed an inverseresponse system. Furthermore, assume that the output disturbance transfer function is Gd(s) = 1/(50s + 1) The performances of four different control schemes are compared: (a) PI controller with inverse-response compensator (IRC), (b) internal model control (IMC), (c) linear quadratic Gaussian (LQG) controller, and (d) multiscale control (MSC). IRC Control Scheme. A PI controller is augmented with the inverse-response compensator (IRC) scheme13 shown in Figure 2. The application of automated tuning using the Matlab SISO Design Tool gives

4. ILLUSTRATIVE EXAMPLES Application of the proposed MSC scheme is demonstrated through the following SISO examples: (1) stable inverseresponse system with one RHP zero, (2) unstable inverseresponse system with one RHP zero, (3) stable inverse-response system with two RHP zeros, (4) integrating inverse-response system with one RHP zero, (5) integrating steam drum level with time delay, and (6) high-order process with time delay. Note that examples 1−4 represent special cases in the sense that their dynamics are multiscale in nature (i.e., the difference in time constants between two successive plant modes is at least 1 order of magnitude). These four examples demonstrate the application (or applicability) of the proposed MSC scheme in a rather ideal situation, as the difference in time constants (dynamics) between two successive plant modes for real processes could often be less than 1 order of magnitude. However, it is important to emphasize that the proposed MSC scheme is applicable to real processes as long as the difference in dynamics between two successive plant modes is significant (might be less than 1 order of magnitude). This is demonstrated by examples 5 and 6. Example 5 demonstrates the applicability/effectiveness of

KPI(s) =

(1.2144s + 0.0264) s

(27)

where the corresponding compensator is given by ⎛ 1 ⎞ 1 ⎟, C(s) = α⎜ − ⎝s + 1 100s + 1 ⎠

α = −0.4

(28)

IMC Scheme. An IMC controller is designed using the Matlab SISO Design Tool based on the nominal model P ̅ (s ) =

−39s + 0.6 100s 2 + 101s + 1

(29)

This leads to the following IMC controller KIMC(s) = F

8.1 × 1015(100s 2 + 101s + 1) 4.988 × 1018s 2 + 5.8 × 1017s + 1

(30)



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Figure 6. Example 1: (a) nominal response and (b) perturbed response (v-gap = 0.2417).

LQG Scheme. An LQG controller is designed based on the nominal model in eq 29 KLQG(s) =

0.016(1.488s 2 + 92.28s + 1) 0.21s 2 + s

outermost control loop. However, for a more complex process, it might be necessary to use a PID controller or some other advanced control laws, such as an LQG controller. Notice from eq 35 that the zeros of the augmented nominal plant Pc occur at s = −1 and 0.0154. Thus, the implementation of subcontroller K1 has no effect on the original location of the RHP zero (at s = 0.0154) corresponding to the nominal plant P̅. In other words, the implementation of the proposed MSC scheme does not cancel out the RHP zero in the sense of the IRC scheme. Recall that, in the IRC scheme, the design principle is to remove the inverse-response dynamics from the feedback loop rather than using this fast factor directly to control the system involved. The performances of the four different control schemes were compared based on 1 unit step change in R and followed by 1 unit step change in Do. For a comparison of performance robustness, the control schemes were tested against 20% modeling errors in both gains and time constants, namely, a0 = −0.2, a1 = 0.2, b0 = 0.2, and b1 = −0.2. These modeling errors lead to v-gap = 0.2417, that is, δ(P̅,P) = 0.2417. Panels a and b of Figure 6 show the nominal and perturbed responses, respectively. Note that all of the control schemes were tuned to provide similar IAE values at the nominal conditions. Based on the IAE values shown in Figure 6b, it is clear that the proposed MSC scheme markedly outperforms the other control schemes, that is, it provides markedly improved performance robustness. Note that the conventional PID controller is unstable at the given perturbed conditions. 4.2. Example 2: Open-Loop Unstable Inverse-Response System. Consider the following second-order, unstable inverse-response system

(31)

MSC Scheme. The procedure given in section 3.3 was used to synthesize the multiscale controller with the aid of the Matlab SISO Design Tool. For this example, the indirect MSC scheme shown in Figure 3b was selected. Thus, the corresponding multiscale predictor was chosen as

W1̅ (s) = P0̅ (s)

(32)

Assuming P = P̅, the inner-layer controller K1 can be designed based on the nominal factor P̅1. For simplicity, it would be recommended to employ the P-only controller for the inner loops. The application of automated tuning (minimum IAE criterion) gives K1 = −40

(33)

Next, by taking n = 1 and substituting eq 33 into eq 18, the closedloop transfer function for the inner layer is obtained as G1(s) =

−40(s + 1) s + 17

(34)

The augmented overall plant transfer function based on eq 29 is Pc(s) = G1(s) P ̅(s) =

1560s 2 + 1536s − 24 100s 3 + 1801s 2 + 1718s + 17

(35)

Finally, the outermost subcontroller K0 is synthesized based on eq 35. The application of automated tuning for PI controller leads to −0.08(95s + 1) K 0(s) = s

P ̅ (s ) = (36)

(38)

where the corresponding plant factors are given by

The overall multiscale controller, which is equivalent to a PID controller with a lag filter, is given by ⎞ 0.188(95s 2 + 96s + 1) ⎛ 1 ⎜ ⎟ K msc(s) = ⎝ s 0.06s + 1 ⎠

−49s + 1.5 100s 2 + 99s − 1

⎤ ⎡ (1 + a0) ⎥ ⎢ ⎡ P0(s)⎤ ⎢ 100(1 + b )s − 1 ⎥ 0 ⎢ ⎥=⎢ ⎥ ⎢⎣ P1(s) ⎥⎦ ⎢ −0.5(1 + a1) ⎥ ⎢⎣ (1 + b1)s + 1 ⎥⎦

(37)

Note that, for simple controller tuning (and to remove steadystate offset), it is often recommended to use a PI controller in the G

(39)



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where the coefficients are kc = 2.19 × 1014, p3 = 186.2, p2 = 348.4, p1 = 162.2, q3 = 9.2 × 1016, q2 = 5.75 × 1016, and q1 = 2.3 × 1016. PID Scheme. A conventional PID controller with a lag filter was tuned based on the IMC tuning available in the Matlab SISO Design Tool

Notice that the RHP zero of the nominal plant (38) occurs at s = 0.0306. Assume that the output disturbance transfer function is Gd(s) = 1/(30s + 1). For comparison purposes, four different control schemes were compared: (a) multiscale control (MSC), (b) LQG controller, (c) IMC controller, and (4) PID controller. MSC Scheme. For the direct MSC scheme shown in Figure 3a, the multiscale predictor was selected as W1(s) = P1̅(s) =

−0.5 s+1

KPID(s) =

(40)

K 0(s) =

(41)

−0.0061(150s + 1) s

(42)

Upon combining eqs 42 and 43 as in eq 24, we obtained the overall multiscale controller K msc(s) =

⎞ 0.0102(149.2s 2 + 150.2s + 1) ⎛ 1 ⎜ ⎟ ⎝ s 0.17s + 1 ⎠

(43)

Note that the controller in eq 44 is the same as a conventional PID augmented with a lag filter. For this example, the zeros of the augmented plant occur at s = −1 and 0.0306. Obviously, the location of the original RHP zero of the nominal plant P̅ remains unchanged. This indicates that there is no dynamics cancellation of the inverse-response component in the feedback loop. LQG Scheme. Using the nominal model in eq 38, an LQG controller can be derived as KLQG(s) =

0.009(170s 2 + 171s + 1) 0.067s 3 + 0.36s 2 + s

⎡ ⎤ 0.8(1 + a0) ⎢ ⎥ ⎡ P (s)⎤ ⎢ 2000(1 + b0)s + 1 ⎥ ⎢ 0 ⎥ ⎢ −0.6(1 + a ) ⎥ 1 ⎥ ⎢ P1(s) ⎥ = ⎢ ⎢ ⎥ ⎢ 100(1 + b1)s + 1 ⎥ ⎥ ⎢⎣ P2(s)⎥⎦ ⎢ ⎢ ⎥ 0.4(1 + a 2) ⎢ ⎥ ⎣ (1 + b2)s + 1 ⎦

(44)

IMC Scheme. The IMC controller was designed based on eq 38, yielding 3

KIMC(s) =

2

q3s + q2s + q1s + 1

(47)

where the nominal plant is given by

2

kc(p3 s + p2 s + p1 s + 1) 3

(46)

To evaluate the performance robustness of the three different control schemes, 20% modeling errors in gains and time constants were introduced, namely, a0 = −0.2, a1 = 0.2, b0 = −0.2, and b1 = 0.2, leading to a v-gap = 0.0796. Panels a and b of Figure show the nominal and perturbed responses, respectively, following sequential step changes of 1 unit each in R, Di, and Do. Notice that the proposed MSC scheme remains stable when subjected to input disturbance Di, thus suggesting that the MSC scheme does not suffer from internal instability. Furthermore, based on the IAE values shown in Figure 7a,b, the proposed MSC scheme can substantially outperform the other control schemes (IMC, LQG, and PID) in terms of nominal performance and performance robustness. Also note that the conventional PID controller is unstable at the given perturbed conditions. 4.3. Example 3: Open-Loop Stable NMP System with Multiple Unstable Zeros. Assume a system with plant factors as follows

Applying the same procedure as used in example 1, we obtained the following two subcontrollers K1 = −10

0.0101( −30.4s 2 + 159.8s + 1) s(1.2s + 1)

P ̅ (s ) =

(45) H

0.394(s 2 − 0.00355s + 7.606 × 10−6) (s + 1)(100s + 1)(2000s + 1)

(48)



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where the coefficients are kc = 1.8 × 10−3, p3 = 1.5 × 105, p2 = 1.52 × 105, p1 = 1.6 × 103, q3 = 1.74 × 104, q2 = 2.57 × 104, and q1 = 140.7. For this example, the perturbed conditions are based on a0 = 0.2, a1 = 0.2, a2 = −0.2, b0 = −0.2, b1 = 0.2, and b2 = 0.2, which gives v-gap = 0.1438. The performances of the three different control schemes were evaluated against sequential step changes of 1 unit each in R and Do. Panels a and b of Figure 8 show the nominal and perturbed responses, respectively. As in example 2, the proposed MSC scheme can be tuned to give superior nominal performance (in terms of the smallest IAE) over the LQG and IMC schemes, yet the MSC scheme can still provide more robust performance than the other two schemes. 4.4. Example 4: Integrating NMP System. Consider an integrating, inverse-response system as follows

Notice that the two RHP zeros in eq 49 occur at s = 0.0018 ± 0.0021j. We assume that the output disturbance transfer function is Gd(s) = 1/(100s + 1). For comparison purposes, the performances of three different control schemes were evaluated: MSC, IMC, and LQG. MSC Control Scheme. By means of the direct MSC scheme shown in Figure 4a, the following three subcontrollers were obtained K 2 = 30

(49)

K1 = −2.2

(50)

K 0(s) =

−(2.16s + 0.0012) s

(51)

Upon combining eqs 51−53, the overall multiscale controller is obtained as ⎡ 1.6(18s 2 + 1.81s + 0.001) ⎤ K msc(s) = ⎢ ⎥Gf (s) s ⎦ ⎣

P ̅ (s ) = (52)

where the high-order filter takes the form Gf (s) =

⎡ ⎤ (1 + a0) ⎢ ⎥ ⎡ P0(s)⎤ ⎢ s[10(1 + b )s + 1] ⎥ 0 ⎢ ⎥=⎢ ⎥ ⎢⎣ P1(s) ⎥⎦ ⎢ −0.7(1 + a1) ⎥ ⎢⎣ 0.3(1 + b1)s + 1 ⎥⎦

(53)

Hence, the overall multiscale controller in eq 52 is equivalent to a conventional PID controller augmented with a second-order filter. IMC Scheme. Based on the nominal model in eq 49, an IMC controller was synthesized as q3s 3 + q2s 2 + q1s + 1

(54)

where the coefficients are kc = 7.04 × 10 , p3 = 2 × 10 , p2 = 2.02 × 105, p1 = 2.10 × 103, q3 = 2.281023, q2 = 7.98 × 1020, and q1 = 5.70 × 1018 LQG Scheme. The following LQG controller was synthesized based on eq 49 15

KLQG(s) =

(57)

Assume that the output disturbance transfer function is Gd(s) = 1/(2s + 1). The zeros of the nominal plant occur at s = −0.408 and 0.351. Thus, the presence of one RHP zero (s = 0.351) confirms that the system exhibits inverse-response behavior. The performances of three different control schemes were compared: MSC, PID, and LQG. MSC Control Scheme. Based on the direct MSC scheme, the subcontrollers are

kc(p3 s 3 + p2 s 2 + p1 s + 1)

3

(56)

where the plant factors are

s+1 (25.4s + 1)(0.075s + 1)

KIMC(s) =

−7s 2 − 0.4s + 1 s(3s 2 + 0.3s + 1)

5

K1 = −12

(58)

2

kc(p3 s + p2 s + p1 s + 1) s(q3s 3 + q2s 2 + q1s + 1)

K 0(s) =

(55) I

−0.0004(370s + 1) s

(59)



Article


Upon approximation of the delay component using the firstorder Padé formula, followed by decomposition, we obtained the nominal factors as

The overall multiscale controller, which is equivalent to PID augmented with a lag filter, is K msc(s) =

5.1 × 10−4(111.1s 2 + 370.7s + 1) s(0.032s + 1)

⎡ 0.037 ⎤ ⎥ ⎡ P0̅ (s)⎤ ⎢ s ⎥ ⎢ ⎥=⎢ ⎢⎣ P1̅(s)⎥⎦ ⎢ −0.037 ⎥ ⎢⎣ 0.5s + 1 ⎥⎦

(60)

PID Scheme. A proportional−integral−derivative (PID) controller augmented with a lag filter was designed using eq 58 through automated tuning based on the minimum IAE criterion (PID-AT) ⎞ 0.006(4.6s + 23.2s + 1) ⎛ 1 ⎜ ⎟ ⎝ s 0.01s + 1 ⎠ 2

KPID1(s) =

Another linearized (perturbed) model at different operating conditions was obtained as

(61)

Another PID controller was designed using the Jeng and Lin tuning formula23 (PID-Jeng) KPID2(s) =

0.0703(s + 0.019) ⎛ 7.764s + 1 ⎞ ⎜ ⎟ ⎝ 0.328s + 1 ⎠ s

PΔ =

(62)

K1 = −30

K 0(s) =

(63)

0.037e−s s

(66)

(67)

−0.55(s 2 + 3s + 0.8) s

(68)

After combining eqs 70 and 71, the overall multiscale controller, which is equivalent to a PID controller augmented with a lead-lag filter, takes the form

The perturbed conditions were based on 30% modeling errors in the gains and time constants, namely, a0 = −0.3, a1 = 0.3, b0 = 0.3, and b1 = −0.3, leading to v-gap = 0.1995. Figure 9a shows the nominal response following sequential step changes of 1 unit in R and −1 unit in Do. The proposed MSC provides the best nominal performance in terms of the smallest IAE value. In addition, the perturbed response is shown in Figure 9b; obviously, the proposed MSC scheme can still outperform the other control schemes including the PID controller designed based on the Jeng and Lin formula23 (PID-Jeng). It is interesting to note that the PID-Jeng scheme can provide improved performance over the PID controller tuned based on the minimization of IAE value (i.e., PID-AT scheme). 4.5. Example 5: Integrating Steam Drum Level. For this example, we employed a typical integrating process with time delay based on the steam drum (SD) level control (from Loop-Pro TRAINER 5.1). A linearized (nominal) model of the SD level is given by P ̅ (s ) =

0.033e−1.64s s

which leads to a v-gap or δ(P̅,PΔ) of 0.0583. With respect to the MSC scheme, the following subcontrollers were obtained

LQG Scheme. An LQG controller was designed as 0.0047(33.66s 3 + 192s 2 + 28.18s + 1) KLQG(s) = 3.36s 3 + 4s 2 + s

(65)

K msc(s) =

26.4(1.25s 2 + 3.75s + 1) ⎛ 0.5s + 1 ⎞ ⎜ ⎟ ⎝ 0.24s + 1 ⎠ s

(69)

Together with the MSC controller in eq 72, a set-point prefilter was chosen as 1/(3s + 1). For comparison purposes, a PID controller was designed using the tuning formula of Lee et al.24 By selecting λ = 1.3, the following PID controller was produced ⎡ 22.3695(0.3132s 2 + s + 0.2506) ⎤ KPID(s) = ⎢ ⎥ s ⎦ ⎣

(70)

which gives a set-point prefilter of 1/(3.6482s + 1). An LQG controller was also designed as KLQG(s) =

(64) J

5.15(0.294s 3 + 1.834s 2 + 4s + 1) 0.0046s 4 + 0.0655s 3 + 0.275s 2 + s

(71)



Article


Figure 11. Example 6: (a) nominal responses and (b) perturbed responses.

A reduced-order model for eq 72 given in Jeng and Lin23 is

and the corresponding set-point prefilter was selected as 1/(3.6s + 1). The control performance was evaluated against 1 unit step change in R followed by −10 unit and −0.5 unit step changes in Di and Do, respectively. Panels a and b of Figure 10 demonstrate the performances of the three different control schemes under nominal and perturbed conditions, respectively. Interestingly, the proposed MSC scheme remains stable when subjected to input disturbance Di, which indicates that it does not suffer from internal instability problems. As in the previous four examples, the proposed MSC scheme can markedly outperform the other control schemes (e.g., conventional PID and LGQ controllers) in terms of nominal performance and performance robustness. 4.6. Example 6: High-Order Process with Time Delay. Consider the high-order process with time-delay reported in Jeng and Lin23 given by P(s) =

( −2s + 1)e−0.5s (2s + 1)(s + 1)(0.5s + 1)

P(̃ s) =

0.94(− 1.62s + 1)e−0.6s (2.71s + 1)(1.50s + 1)

(73)

Using the reduced-order model in eq 73, an overall multiscale controller was obtained based on the three-layer direct MSC as K msc(s) =

0.35(s + 1.22)(s + 0.43) Gf (s) s

(74)

where the second-order lead-lag filter is expressed as Gf (s) =

(0.3s + 1)(1.5s + 1) (0.16s + 1)(0.86s + 1)

(75)

Figure 11 compares the performances of the proposed MSC controller and the conventional PID controller designed based on the Jeng and Lin formula.23 The two controllers exhibit similar nominal performances in terms of IAE values (Figure 11a). However, in the presence of 15% modeling errors (i.e., 15% error in time delay, 15% error in gain, and −15% error in time constant with respect to the original model), the proposed MSC controller

(72) K



Article

(5) Pedret, C.; Vilanova, R.; Moreno, R.; Serra, I. A Refinement Procedure for PID Controller Tuning. Comput. Chem. Eng. 2002, 26, 903−908. (6) Scali, C.; Rachid, A. Analytical Design of Proportional−Integral− Derivative Controllers for Inverse Response Processes. Ind. Eng. Chem. Res. 1998, 37, 1372−1379. (7) Chien, I. L.; Chung, Y. C.; Chen, B. S.; Chuang, C. Y. Simple PID Controller Tuning Method for Processes with Inverse Response Plus Dead Time or Large Overshoot Response Plus Dead Time. Ind. Eng. Chem. Res. 2003, 42, 4461−4477. (8) Gu, D.; Ou, L.; Wang, P.; Zhang, W. Relay Feedback Auto Tuning Method for Integrating Processes with Inverse Response and Time Delay. Ind. Eng. Chem. Res. 2006, 45, 3119−3132. (9) Zhang, Y.; Wang, W.; Wang, J. S. Optimal Design of PI/PD Controller for Non-Minimum Phase System. Trans. Inst. Meas. Control 2006, 28, 27−35. (10) Sanchis, R.; Romero, J. A.; Balaguer, P. Tuning of PID Controllers based on Simplified Single Parameter Optimization. Int. J. Control 2010, 83, 1785−1798. (11) Stephanopoulos, G. Chemical Process Control: An Introduction to Theory and Practice; Prentice-Hall: Englewood Cliffs, NJ, 1984. (12) Smith, O. J. M. A Controller to Overcome Dead Time. ISA J. 1959, 6, 28−33. (13) Iinoia, K.; Alperter, R. J. Inverse Response in Process Control. Ind. Eng. Chem. 1962, 54, 39−43. (14) Alcantara, S.; Vilanova, R.; Zhang, W. D. Analytical H∞ Design for a Smith-Type Inverse-Response Compensator. In Proceedings of the American Control Conference; IEEE Press: Piscataway, NJ, 2009; pp 1604−1609. (15) Nasution, A. A.; Jeng, J. C.; Huang, H. P. A Simple Non-Minimum Phase Zero Predictor for Open-Loop Unstable Processes. Asia-Pac. J. Chem. Eng. 2011, 6, 441−451. (16) Wang, Q. G.; Bi, Q.; Zhang, Y. Partial Internal Model Control. IEEE Trans. Ind. Electron. 2001, 48, 976−982. (17) Uma, S.; Chidambaram, M.; Rao, A. S. Set Point Weighted Modified Smith Predictor with PID Filter Controllers for NonMinimum-Phase (NMP) Integrating Processes. Chem. Eng. Res. Des. 2010, 88, 592−601. (18) Morari, M.; Zafirou, E. Robust Process Control. Prentice-Hall International: Englewood Cliffs, NJ, 1989. (19) Abe, N.; Yamanaka, K. Smith Predictor Control and Internal Model ControlA Tutorial. In SICE Annual Conference, Fukui, Japan, August 4−6, 2003; IEEE Press: Piscataway, NJ, 2003; pp 1383−1387. (20) Garcia, P.; Santos, T.; Normey-Rico, J. E.; Albertos, P. Smith Predictor-Based Control Schemes for Dead-Time Unstable Cascade Processes. Ind. Eng. Chem. Res. 2010, 49, 11471−11481. (21) Vinnicombe, G. Uncertainty and Feedback: H∞ Loop Shaping and ν-Gap Metric; Imperial College Press: London, 1999. (22) Anderson, B. D. O.; Brinsmead, T. S.; Bruyne, F. D.; Hespanha, J.; Liberzon, D.; Morse, A. S. Multiple Model Adaptive Control. Part 1: Finite Controller Coverings. Int. J. Robust Nonlinear Control 2000, 10, 909−929. (23) Jeng, J. C.; Lin, S. W. Robust Proportional−Integral−Derivative Controller Design for Stable/Integrating Processes with Inverse Response and Time Delay. Ind. Eng. Chem. Res. 2012, 51, 2652−2665. (24) Lee, Y.; Lee, J.; Park, S. PID Controller Tuning for Integrating and Unstable Processes with Time Delay. Chem. Eng. Sci. 2000, 55, 3481− 3493.

again shows markedly better performance robustness than the conventional PID controller (Figure 11b). Note that this example shows that the MSC controller can also be designed based on a reduced-order model for a case of highorder processes. It is interesting to point out that the designed MSC controller is actually equivalent to a PID controller augmented with a filter. Thus, the proposed MSC scheme can be used as an alternative method to design a PID controller that often provides better performance than a PID controller designed by some conventional methods.

5. CONCLUSIONS A novel multiscale control (MSC) scheme with a general design procedure has been presented in this article. This scheme is rather versatile in application, as it can be used to effectively control either stable or integrating/unstable nonminimum-phase (NMP) processes including those with time delays. In principle, the proposed control scheme aims to achieve good cooperation among the different plant modes. We believe that this is vital to achieving good control performance and performance robustness. Good cooperation among the different plant modes can be accomplished through a cascaded combination of several multiscale subcontrollers, where each subcontroller is designed to control a specific plant mode. The applicability and effectiveness of the MSC scheme have been demonstrated through a number of single-input−single-output (SISO) examples. Based on an extensive numerical study, it was shown that the proposed MSC scheme is capable of achieving nominal performance and performance robustness superior to those of some well-established control schemes (e.g., PID, IMC, and LQG controllers). It is worth highlighting that the proposed MSC scheme can also provide a powerful method to synthesize a PID controller augmented with a filter, where the filter can be of a high-order form. It is interesting to note that the idea of multiscale control described in this article represents a new paradigm of controller synthesis for complex processes. As such, the idea might lead to new research opportunities in both academia and industry. Possible interesting research projects might include the design of a multiscale control scheme for linear multi-input−multi-output (MIMO) systems, the development of a multiscale control scheme incorporating an iterative identification-controller robustness refinement algorithm for application to nonlinear systems, the development of a multiscale control scheme for cascade control systems, and the systematic evaluation of the stability and robustness properties of the proposed multiscale control scheme.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +6085-443824. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Åström, K. J. Limitations on Control System Performance. Eur. J. Control 2000, 6, 2−20. (2) Horowitz, I.; Liao, Y. K. Limitations of Non-Minimum-Phase Feedback Systems. Int. J. Control 1984, 40, 1003−1013. (3) Qiu, L.; Davison, E. J. Performance Limitations of Non-Minimum Phase Systems in the Servomechanism Problem. Automatica 1993, 29, 337−349. (4) Kumar, R.; Moore, J. B. On Adaptive Minimum Variance Regulation for Non-Minimum Phase Plants. Automatica 1983, 19, 449−451. L


Novel Multiscale Control Scheme for Nonminimum-Phase Processes

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