Optimal H2 Input Load Disturbance Rejection Controller Design for

Article pubs.acs.org/IECR

Optimal H2 Input Load Disturbance Rejection Controller Design for Nonminimum Phase Systems Based on Algebraic Theory Bo Sun,† Wei Zhang,† Weidong Zhang,*,† and Zhijun Li‡ †

Key Laboratory of System Control and Information Processing, Ministry of Education of China, and Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China ‡ College of Automation Science and Engineering, South China University of Technology, Guangzhou 510006, People’s Republic of China S Supporting Information *

ABSTRACT: This work discusses the issue of input load disturbance rejection (ILDR) for open-loop nonminimum phase (NMP) plants. A novel analytical solution is proposed on the basis of the internal model control (IMC) theory. Differing from other methods, the proposed design is conducted to optimize the ILDR criterion. Optimization of the input disturbance response of the controller is performed under the constraints on robustness. When the input load disturbance is taken into consideration, the proposed controller performs better disturbance rejection capability in terms of 2-norm than most explored IMC-based controllers derived from the conventional criterion. Typical NMP processes are systematically analyzed. Numerical examples are given to illustrate the effectiveness of the novel solution. The quantitative performance specifications and robust stability can be obtained by monotonously tuning the single parameter. Results show that the proposed solution makes the proposed method yield the expected dynamic responses.

1. INTRODUCTION The issues of load disturbance rejection have been considered as one of the most significant aspects in industrial practice.1 Since the existence of unexpected load disturbance would destroy the balance status of a plant that has been appropriately set up, a great deal of attention was paid to both theory and applications.2 The processes, which contain right half-plane zero(s) or time delay, are called nonminimum phase (NMP) systems due to their specific phase response characteristics. Such systems are exhibited by a number of processing units, such as drum boiler and distillation column.3 For NMP plants, the synthesis procedure becomes more complex and challenging. The constraints, such as bandwidth limitation and achievable sensitivity reduction, result in its rigorous internal stability conditions and vulnerability against external disturbances.4,33 Over the past decades, the IMC theory has been widely recognized as one of the most effective strategies for disturbance rejection and has been successfully applied for different types of cases.5 A number of works providing control schemes or tuning strategies in terms of the IMC principle were proposed to optimally reject load disturbance.6−8 With a focus on the improvement of the closed-loop disturbance rejection performance, IMC-based proportion-integral-derivative (PID) tuning strategies for open-loop stable and unstable plants were elaborated.9−11 With the Pade approximation approach, analytical solutions are developed12−14 and then the corresponding PI/PID tuning formulas were given for different linear time-invariant processes. By applying the Smith predictor structure, Tian and Gao15 improved the load disturbance rejection capability for integrating processes with dominant time delay. Several Smith predictor-based control schemes aiming to enhance the load disturbance rejection ability were proposed.16−19 Owing to the superiority of the Smith predictor © 2013 American Chemical Society

structure for the load disturbance rejection objective, a variety of modified Smith predictor control schemes were also derived for open-loop unstable plants.20−22 Because the Smith predictor structure is in essence equivalent to the IMC structure, alternative two-degrees-of-freedom (2DOF) IMC-based schemes were presented as well to further improve disturbance rejection performance.23,24 Recently, Normey-Rico and Camacho proposed a unified approach for robust dead-time compensator design based on the filtered Smith predictor structure.25 With the proposed structure, controller tuning procedures for integrative and unstable processes are analyzed. In light of the deficiencies in existing IMC-based methods for load disturbance rejection with slow dynamics, modified IMCbased control strategies were proposed for integrating and unstable processes to enhance the performance of disturbance response.26,27 For further improvement on the load disturbance rejection, the systems that suffer both input and output external load disturbance are also taken into consideration. By appropriately selecting the weighting function, Alcantara et al.28,29 demonstrated the IMC-based control scheme for balancing input/output disturbance response. Notice that most of the existing methods for load disturbance rejection were devoted to optimize the 2-norm performance objective min∥W(s) S(s)∥2,25, where S(s) denotes the closedloop sensitivity function and W(s) is a user-specified weighting function. How to design the optimal controller for the ILDR is still an open question. The objective of this issue is to optimize the 2-norm criterion min∥W(s) G(s) S(s)∥2. In most cases, the Received: Revised: Accepted: Published: 1515

August 15, 2013 November 15, 2013 December 22, 2013 December 23, 2013 dx.doi.org/10.1021/ie402682t | Ind. Eng. Chem. Res. 2014, 53, 1515−1528

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematics of the (a) classical feedback control loop and (b) the IMC structure.

right half-plane. It is assumed that N− (0) = N+ (0) = M− (0) = M+ (0) = 1, and the degree condition satisfies

ILDR problem is neglected for its complexity and the tedious design procedure. However, the input load disturbance widely exists in industrial and chemical processes. The aim of this issue is to provide optimal analytical solution for single-input/singleoutput (SISO) NMP plants to improve ILDR response. For clarity and convenience, the solution is analyzed in the unity feedback framework. It can also be extended to other control structures. The outline of the rest part of this work is organized as follows: In section 2, essential background materials are reviewed first, including controller parametrization and optimal H2 controller design based on algebraic theory. The issue to be discussed is clarified. In section 3, a novel controller design strategy for the ILDR is introduced on the basis of the algebraic theory and the systematic proof process is given, which is the main contribution of this work. In section 4, the proposed method is discussed for several typical stable processes. Theoretical analysis is given, and rigorous results are derived. In section 5, numerical examples are given to illustrate the superiority of the proposed method. Finally, section 6 summarizes the main ideas and makes the concluding remarks.

deg{N −(s)} + deg{N+(s)} ≤ deg{M −(s)} + deg{M+(s)} (2-3)

This assumption guarantees that the plant is proper. Notice that the defined description includes a wide range of plants which means that the proposed result is suitable for almost all the cases, especially for the plants with RHP zeros. Due to the constraints caused by the specific phase response characteristics, many design methods cannot treat them directly. Therefore, an approximated model is usually adopted for design. Different from them, the proposed method can directly treat the plants with RHP zeros without approximation. 2.2. Controller Parametrization. The closed-loop system in Figure 1a is internally stable if and only if all elements in the following transfer function matrix are stable: ⎡ C(s)G(s) ⎤ G (s ) ⎢ ⎥ ⎢ 1 + C(s)G(s) 1 + C(s)G(s) ⎥ H (s ) = ⎢ ⎥ C(s) C(s)G(s) ⎥ ⎢ − ⎢⎣ 1 + C(s)G(s) 1 + C(s)G(s) ⎥⎦

2. BACKGROUND MATERIALS AND PROBLEM DESCRIPTION In this section, preliminary knowledge is presented. Most of the assumptions and definitions used in this work are adopted from ref 13. 2.1. Background Materials. The block diagram of the classical feedback and IMC control structure is shown in Figure 1, where G(s) is the real plant, Gm(s) is the nominal model of G(s), C(s) is the linear time-variant controller to be designed, and Q(s) is the IMC controller. r(s), di(s), do(s), u(s), and y(s) denote the reference input, input disturbance, output disturbance, control signal, and process output, respectively. Gm(s) herein is described as KN+(s)N −(s) −θs G (s ) = e M+(s)M −(s)

where Y (s) = H(s)X(s);

i

i=1

∀ τi , τj > 0

S(s ) =

1 1 + C(s)G(s)

(2-6)

T (s ) =

C(s)G(s) 1 + C(s)G(s)

(2-7)

ns + nu

∏ (τis + 1)k ;

M+(s) =

∏

(2-5)

Assuming that the model is exact, the sensitivity transfer function (i.e., the transfer function from the set-point r (s) to the error e (s)) and the complementary transfer function (i.e., the transfer function from the set-point r (s) to the system output y (s)) are given by

where M −(s) =

Y (s) = [ y(s) u(s)]T ;

X(s) = [ r(s) d i(s)]T

(2-1)

ns

(2-4)

( −τjs + 1)kj

The effect of the unity feedback controller C(s) on the sensitivity function and the complementary sensitivity function is very complicated. If one defines the transfer function Q(s) as

j = ns + 1

(2-2)

Q (s ) =

Here K is a real constant that denotes the static gain, and θ is a positive real constant denoting the pure time delay. The subscript minus sign (−) denotes that the roots are in the left half-plane (LHP) and the subscript plus sign (+) denotes that the roots are in the closed right half-plane (RHP); that is, N− (s) and M− (s) are polynomials with roots in the left half-plane, and N+ (s) and M+ (s) polynomials with roots in the closed

C(s) 1 + C(s)G(s)

(2-8)

it follows that S(s ) = 1 − G (s )Q (s )

T (s ) = G (s )Q (s ) 1516

(2-9) (2-10)

dx.doi.org/10.1021/ie402682t | Ind. Eng. Chem. Res. 2014, 53, 1515−1528


Article

The effect of the controller Q(s) on the sensitivity function and the complementary sensitivity function is direct. With respect to the above derivation, eq 2-4 is re-formed as ⎡G(s)Q (s) [1 − G(s)Q (s)]G(s)⎤ ⎥ H (s ) = ⎢ ⎥⎦ ⎢⎣Q (s) − G (s )Q (s )

disturbance rejection problem, we can obtain the optimal controller that makes the closed-loop system simultaneously achieve the optimal set-point tracking performance and the load disturbance rejection capability, because the reference tracking error and the load disturbance have the same transfer function, the design for the regulator problem is identical to that for the servo problem: r(s) = do (s) = 1/s. For the input load disturbance, the obtained controller may achieve only one target. To separately optimize the system performance of the input load disturbance without sacrificing the nominal set-point tracking performance, both the standard and many modified 2DOF control structures can be utilized to decouple the setpoint tracking and ILDR and to simultaneously achieve the optimal set-point tracking and ILDR.

(2-11)

According to ref 13, the following theorem is given. Lemma 2.1. Let G(s) be a plant with a time delay. The unity feedback system is internally stable if and only if (1) Q(s) is stable, (2) the ration function 1−G(s)Q(s) has zeros wherever G(s) has unstable poles, and (3) there is no closed right halfplane zero-pole cancellation in C(s). Reference 13 has provided a systematic analysis of the timedelay system optimal controllers design issues. First, the parametrization is given as follows: Lemma 2.2.13 Assume that G(s) is a SISO plant with time delay. All controllers that make the unity feedback control system internally stable and have a zero steady-state error for a step reference can be parametrized as

C(s) =

Q (s ) 1 − G (s ) Q ( s )

3. OPTIMAL ILDR CONTROLLER DESIGN In this section, the analytical solution for the ILDR controller is derived first. Then, we specifically discuss several typical NMP processes. Finally, we analyze the nominal performance, robust stability, and robust performance properties. 3.1. Optimal H2Controller Design Procedure. The subject of this subsection is to design the H2 optimal controller for the plant with time delay. The performance criterion is given in eq 2-16. Although the following design procedure is sufficiently general for ramp or more-complex inputs, we only consider step inputs for clarity. In this case we can take W(s) = 1/s. The control system design problem can be considered as a search or an optimization over the set of all stabilizing controllers with asymptotic properties. For the H2optimal performance criterion, we have: Theorem 3.1. Let G(s) be a plant in the form of eqs 2-1 and 2-2; the optimal H2 controller Copt(s) for the input load disturbance is given as

(2-12)

where Q (s ) =

[1 + sQ 2(s)]M+(s) (2-13)

K

Q2(s) is any stable transfer function that makes Q(s) proper and satisfies lim

s → 1/ τj

[1 + sQ 2(s)]N+(s)N −(s)e−θs ⎫ ⎪ ⎪ dk ⎧ ⎨ ⎬ − =0 1 k⎪ ⎪ M −(s) ds ⎩ ⎭

for j = ns + 1, ..., ns + nu ;

0 ≤ k < kj

(2-14)

and there is no closed right half-plane zero-pole cancellation in C(s). Before stating the main result, we present another essential lemma referred to in the following. Lemma 2.332 If F1 ∈ H2 and F2 ∈ H⊥2 , the following equation is satisfied: || F1 + F2 ||2 2 = || F1 ||2 2 + || F2 ||2 2

Copt(s) =

Q opt(s) =

(2-15)

(3-1)

(sY (s) + 1) M+(s)M −(s) N+( −s)N −(s) KN+( −s)N −(s)

(3-2)

Y (s) is a rational polynomial and satisfies the following conditions. condition 1: lim

s →−1/ τi

dk (N −(s)N+2( − s)e θs − N+(s) − sN+(s)Y (s)) = 0 k ds

for ∀ i = [1, ns],

k = 1, 2, ..., ki

condition 2: lim

s →−1/ τj

(2-16)

dk (N −(s)N+2( − s)e θs − N+(s) − sN+(s)Y (s)) = 0 ds k

for ∀ j = [ns + 1, ns + nu],

Notice that the transfer function from the input load disturbance to the output is G (s ) Gdo(s) = 1 + G(s)C(s)

1 − G(s)Q opt(s)

where

Based on the aforementioned statements, Zhang et al.13 has proposed the optimal result. However, the work does not provide an analytical solution for the input disturbance rejection problem. This work aims to solve the aforementioned problem. The problem can be stated as follows: with respect to the NMP plant G(s) in the unity feedback loop depicted in Figure 1a, find the internally stabilized controller C(s) to optimally decrease the input load disturbance effect based on the optimal performance criterion; min || W (s)G(s)S(s)||2

Q opt(s)

k = 1, 2, ..., ki

condition 3: Y (s) does not contain prediction

(2-17)

condition 4:

which is not the same as that from the reference input r(s) to the error signal e(s). When considering the output load

Y (s) does not contain any poles 1517



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class of processes, the following theorem gives the explicit solution for this issue. Theorem 3.2. Let G(s) be a plant in the form of eq 2-1. Assume G(s) does not contain multiplicity poles and satisfies τi ≠ τj; ∀i ∈ [1, ns], ∀j ∈ [ns + 1, ns + nu], and the unique optimal H2 controller Copt(s) for the plant against input load disturbance is given as

condition 5: the order of Y (s) is less than deg {M+(s)} + deg {M+(s)}

Proof: Substituting the plant into the performance criterion, we have || W (s)G(s)S(s)||22 N (s)N −(s) −θs 1 KN+(s)N −(s) −θs⎡ e ⎢1 − + e ⎣ s M+(s)M −(s) M −(s)

=

⎤ sN+(s)N −(s) −θs e Q 2(s)⎥ ⎦ M −(s)

−

Copt(s) =

2

2

Q opt(s) 1 − G(s)Q opt(s)

(3-8)

where

In light of the definition of 2-norm, an all-pass portion does not affect the value of its 2-norm. The time delay is all-pass. The following rational transfer function is also all-pass

Q opt(s) =

(sY (s) + 1) M+(s)M −(s) N+( −s)N −(s) KN+( −s)N −(s)

(3-9)

2

Gall‐pass(s) =

N+ (s)M+( −s) N+2( −s)M+(s)

ns + nu

(3-3)

Y (s ) =

Therefore,

−

N+ (s)N − (s) M+(s)M −2(s)

e−θsQ 2(s) (3-4)

2

|| W (s)G(s)S(s)||22 N −(s)N+2( −s)e θs − N+(s) sM+( −s)M −(s)N+(s)

= K2 + −

M −(s) − N −2( −s)N+2( −s) sM+( −s)M −2(s) N+2( −s)N −2(s) M+( −s)M −2(s)

2

Q 2(s) (3-5)

2

|| W (s)G(s)S(s)||22 = K 2 || G+(s)||22 + K 2 || G−(s)||22

(3-6)

ns + nu

where G+(s) =

τa(τbs + 1) ⎫ ⎬ τa − τb ⎭ ⎪

⎪

(3-10)

Proof: Equation 3-6 shows that X(s) should be selected so that G+(s) and G− (s) are orthogonal according to lemma 2.1. Thus, X(s) possesses the following properties: (1) sM+(−s)M−(s) is a factor of N−(s)N2+(−s)eθs − N+(s) − sX(s); (2) N+(s) is a factor of X(s). When the proper X (s) that satisfies the above two conditions is obtained, the rational fraction G+(s) only contains right-half-plane poles and G− (s) only has left-half-plane poles. If we let G−(s) = 0, we can get Q2(s). The following analysis tells that the satisfied Q2(s) is unique. To satisfy the aforementioned condition (2), we define Q2(s) as eq 3-6. Without a loss of generality, all of the Y (s)s that satisfy this condition for the class of processes can be expressed as

2

2

∏ b = 1, b ≠ a

sM+(s)M −2(s) 2

⎪

ns + nu

×

M −(s)N+(s)N −(s) − N+2(s)N −2(s)e−θs

2

⎧ N 2(1/τ )N ( −1/τ )e−θ / τa − N ( −1/τ ) a a a − + ⎨ + ( −1/τa)N+( −1/τa) ⎩ ⎪

a=1

|| W (s)G(s)S(s)||22 =K

∑

Y (s ) =

∑ a=1

N −(s)N+2( −s)e θs

− N+(s) − sX(s) sM+( −s)M −(s)N+(s)

⎧ N 2(1/τ )N ( −1/τ )e−θ / τa − N ( −1/τ ) − + a a a ⎨ + − τ − τ ( 1/ ) N ( 1/ ) ⎩ a + a ⎪

⎪

ns + nu

× [1 + (τas + 1)Ka(s)] ∏ b = 1, b ≠ a 2

G−(s) =

2

M (s) − N+ ( −s)N − (s) X (s ) + − M+( −s)M −(s)N+(s) sM+( −s)M −2(s) −

N+2( −s)N −2(s) M+( −s)M −2(s)

X(s) = N+(s)Y (s)

τa(τbs + 1) ⎫ ⎬ τb − τa ⎭ ⎪

⎪

(3-11)

where

Q 2(s)

Ka(s) = sKa1(s)

(3-12)

and Ka1(s) is a stable rational transfer function. According to the definition of 2-norm,33 the order condition should be satisfied such that the remaining part of the criterion should be a strictly proper rational transfer function. The order of the denominator should be not lower than that of the numerator. The remaining part of the criterion is

(3-7)

Y(s) satisfies conditions (1), (2) and (4). By minimizing G−(s), i.e., letting it equal zero, we can obtain the optimal controller as eqs 3-1 and 3-2. According to the definition of 2-norm, the normed transfer function should be strictly proper and causal. The controller needs to meet conditions (3) and (5). For a 1518



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Figure 2. (a) Surface of the ratio of ISE1/ISE2 and (b) relationship between α and the ratio.

Notice that ref 13 has discussed the same question and given an example, which is a special case of the proposed solution. To illustrate the performance of the problem, consider the simplest NMP process described by the following transfer function:

|| W (s)G(s)S(s)||22 = K 2 N −(s)N+2(− s)e θs − N+(s) − sN+(s) ⎧ N 2(1/τ )N (− 1/τ )e−θ / τa − N (− 1/τ ) a − a + a ⎨ + (− 1/τa)N+(− 1/τa) a=1 ⎩ ns + nu τ (τ s + 1) ⎫ ⎬ × [1 + (τas + 1)Ka(s)] ∏ a b τb − τa ⎭ b = 1, b ≠ a ns + nu

∑

×

⎪

G (s ) = K

⎪

⎪

2

ISE1 = min || W (s)G(s)S(s)||22

/[sM+(− s)M −(s)N+(s)] 2

deg{den} = 1 + deg{M+(s)} + deg{M −(s)} + deg{N+(s)}

2

=

(3-13)

and the order of the numerator is

+ deg{M −(s)} − 1 + deg{Ka(s)}

a0 =

(3-14)

K i(s) = 0

2α τs + 1

⎪

∏

τa(τbs + 1) ⎫ ⎬ τa − τb ⎭

=

⎪

⎪

θs

N −(s)N+ ( −s)e − N+(s) − sX(s) sM+( −s)M −(s)N+(s)

2

2K α τ

(3-21)

To compare the two performances, we calculate the ratio of them

(3-16)

τa 2 ISE1 4ατ = 03 = θ and (b) α < θ).

Figure 12. Disturbance response of model mismatch in (a) α and (b) τ.

quickly, which means it has a small transient process time. The transient time increases with the increase of λ/τ value. But, the disturbance response holds a large undershoot value. Contrary to the transient time, the undershoot value decreases with the increase of λ/τ . At the beginning, the undershoot value dominates the whole performance, resulting in that value decreasing initially with the increasing λ/τ for different θ/τ ratios. At a specific λ value, the transient process time and undershoot value have a balancing effect on the performance. After that value, the transient process time, instead of the undershoot value, dominates the performance. That is the reason behind the above observation. We also study the relationship between θ/τ (sweeping over the ratio range from 0.01 to 5) and the performance value at fixed robust level (Ms = 1.8 and 1.6). The proposed result is compared with the result in ref 9. The results are shown in Figure 10. The result shows that for both of the cases α > θ and α < θ, the proposed method performs a better disturbance rejection response for the smaller ISE value. The optimal performance of the optimal ILDR controller has been proved better than that of the conventional one. The augmented

The set-point tracking response is shown in Figure 8. It is clear that, under the 1DOF control structure, the proposed method performs an excessive overshoot in the set-point tracking response. This method designed specifically for the ILDR issue is not the optimal choice for set-point tracking. Therefore, to simultaneously obtain good set-point tracking and disturbance rejection responses, a 2DOF control structure is suggested. Since the proposed IMC-based PID tuning rules have a single tuning parameter, it is closely related to the closed-loop performance as well as the robustness of the system. Therefore, it is important to analyze the effect of λ on the performance values. The IAE and ITAE of the proposed method for different λ/τ values are calculated for the cases of θ/τ = 0.5, 1.0, 1.5, and 2.0. The corresponding results are shown in Figure 9a and Figure 10b. For each case, the IAE and ITAE value decreases initially with the increasing λ/τ for different θ/τ ratios. As the θ/τ ratio increases from 0.5 to 2.0, the variation of ISE comparatively increases. The trend of ISE decreasing with λ reverses after a specific λ value for each θ/τ ratio. For a small value of λ/τ, the system can eliminate the disturbance effect 1525



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Figure 13. Disturbance response of model mismatch in (a) θ and (b) K.

Figure 14. Disturbance response of the worst-case model mismatch.

Figure 15. Input disturbance response for G(s) = (−s + 1)e−2s/(2s + 1) (3s + 1).

with the increase of θ/τ . The results also show that, with the increase of θ/τ, λ become less sensitive to the robust level values (Ms) since they have small changes when θ/τ > 2.

controller also follows this point. In Figure 11, the relationship between θ/τ and λ is studied at a fixed system robust level. The result shows that for both of the methods, the λ values decrease 1526



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Table 2. Controller Parameter and Performance Indices for Example G(s) = (−s + 1)e−0.5s/(2s + 1)(3s + 1) input disturbance rejection

set-point tracking response

filter

λ/θ

ISE

IAE

ITAE

TV

ISE

IAE

ITAE

TV

proposed

(sY + 1)(βs + 1) (αs + 1)(λs + 1)3

3.8

0.942

2.868

19.547

1.634

7.850

8.483

42.667

12.41

Horn et al.34

βs + 1 (λs + 1)2

4.0

0.963

2.935

21.154

1.693

7.953

8.760

45.870

10.20

Lee et al.36

β1s 2 + β2s + 1

3.8

0.962

2.900

20.061

1.686

7.941

8.740

45.720

10.21

tuning methods

4

(λs + 1) Shamsuzzoha and Lee9

(βs + 1)2 (λs + 1)3

4.0

0.953

2.849

19.394

1.713

7.953

8.789

46.356

10.20

Morari and Zafiriou1

1 λs + 1

2.2

1.019

3.196

24.011

1.54

7.882

8.485

42.401

10.68

Lee et al.35

β1s 2 + β2s + 1

4.0

0.963

2.935

21.154

1.69

7.953

8.760

45.870

10.20

2

(λs + 1)

Although numerous works have been developed for the improvement of the ILDR performance, most of them are trying to provide effective filters. The optimal formula proposed in the paper for the ILDR issue has not been explicitly presented. It has been proved that the optimal ILDR performance can be improved when the proposed result is applied. When the augmented optimal controller is considered, the simulations show that, for the system with fixed robust levels, the proposed method performs better disturbance rejection response for the smaller performance criterion values. In conclusion, the presented optimal ILDR controller can be used as a base function for the ILDR issue and can be augmented by other filters to further improve the ILDR performance.

The robustness of the controller is evaluated by imposing a perturbation uncertainty of 20% in all parameters. The simulation results are given in Figures 12−14. The results show that the model mismatch of parameters does not affect the final condition of the disturbance response, that the effect of load disturbance would be asymptotically eliminated, and that the closed-loop system has no steady-state error. 5.2. Example 2: Second-Order NMP Process. In this section, an example is given for a second-order NMP process. The process is depicted as G (s ) =

−s + 1 e−0.5s (2s + 1)(3s + 1)

(5-2)

■

For the first simulation, the unit step set-point input begins at t = 5 s. A unit load disturbance is imposed on the system at t = 50 s. The corresponding disturbance rejection response is shown in Figure 15. The performance specification for the ILDR is listed in Table 2. For fair comparison, all λ are tuned with the same closed-loop system robust level (Ms = 1.78). The performance matrix shows that the disturbance rejection responses provided by the proposed method and ref 9 are better than those of the others since they generate smaller values of ISE, IAE, and ITAE specifications. But, the same as the condition appearing in the first simulation, the closed-loop system of the proposed method generates the maximum TV, which is a good measure of the smoothness of the disturbance rejection response and the controller output, respectively. On the basis of the discussion above, it is concluded that the proposed method has the best disturbance rejection performance (measured by ISE) at the cost of sacrificing the smoothness of both the disturbance rejection response and the controller output.

ASSOCIATED CONTENT

S Supporting Information *

Equations S1−S3 and Table S1. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest

■

ACKNOWLEDGMENTS The work is partly supported by the National Science Foundation of China under Grants 61025016, 11072144, 61034008, and 61221003.

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REFERENCES

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6. CONCLUSION In this work, we give a treatment for the optimal ILDR controller design problem in a quadratic cost setting. The proposed design is conducted to optimize the ILDR criterion. A novel analytical solution is proposed on the basis of the internal model control (IMC) theory. Besides, the relationship between the proposed result and the conventional IMC one has been analyzed. Optimization of the input disturbance response of the controller is performed under the constraints on robustness. 1527



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Optimal H2 Input Load Disturbance Rejection Controller Design for

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