Robust fast controller design via nonlinear fractional differential equations

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Robust fast controller design via nonlinear fractional differential equations Xi Zhou, Yiheng Wei, Shu Liang, Yong Wang n Department of Automation, University of Science and Technology of China, Hefei 230027, China

art ic l e i nf o

a b s t r a c t

Article history: Received 2 July 2016 Received in revised form 15 February 2017 Accepted 17 March 2017

A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Fractional order systems Nonlinear control Robustness Fast response

1. Introduction Overshoot and rise time of system output are two significant engineering indices to evaluate a control system performance. However, these two indices are usually conflicting with each other, which is a well-known fact. Especially, for linear plants with PID controllers, when the overshoot is compensated to a small value by tuning the PID parameters, the rise time of system response will certainly become slow, and vice versa. In fact, such phenomenon is caused by the inherence limitation of controller design, which is therefore impossible to break through by tuning the controller parameters only. Most of the existing methods trade off these two indices in order to guarantee the feasibility of a traditional controller. In view of above discussions, a nature question is that can we break through the limitation of traditional methods and develop an advanced controller that improves both the overshoot and rise time performance? This paper is denoted to this challenging problem. In fact, many existing advanced controllers have appealing properties beyond traditional ones. To effectively promote the control system performance, nonlinear switching control method has been proposed and widely applied in control theories and engineering practices [1,2]. In particular, the well known bang-bang controller renders the closed-loop system convergence in finite time [3], that is to say, possessing fast response performance, and can be easily implemented. Meanwhile, the sliding-mode control using signed power function as feedback is recognized as one of the efficient tools to n

Corresponding author. E-mail address: [email protected] (Y. Wang).

design robust controllers for complex high-order nonlinear dynamic plant operating under uncertainty conditions [4,5]. On the other hand, the fractional order systems (FOSs) have attracted lots of attention during the past few years since many engineering plants and processes that cannot be more accurately described without the introduction of fractional order calculus [6,7]. As a result of the tremendous efforts devoted by researchers, the modeling [7,8], stability analysis [9,10], controller design [11– 13] and numerical approximation method [14] and so on, now involve FOSs. For robust controller design, Oustaloup has proposed the famous CRONE (Commande Robuste d'Ordre Non Entier) methodology [15], which provides a frequency-domain approach for the design of output feedback robust controllers for both integer and fractional order LTI systems. It has the reference model as Bode's ideal loop transfer function, while the purpose of CRONE controller design is to obtain an open-loop characteristic similar to that of this reference model. It was used in [16] to solve the speed control problem of multi-mass systems, while the controller permitted to ensure the robust speed control of the load in spite of plant parametric variations and speed observation errors. Morand et al. [17] had dealt with car longitudinal control performed by cruise control system, which had performed much better results in terms of robustness to mass and velocity uncertainties than the classical PI control method. The CRONE approach was also used in [18] to control the temperature of a diffusive medium. The stability robustness was guaranteed despite of variations of open-loop gain from the parametric uncertainties. However, almost all the works are limited to the linear system framework. Moreover, as limited to the inherent attributes of such linear ideal loop transfer function,

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it is difficult to improve the other index, such as the system rise time, by adjusting the controller parameters. Thus, all those researches focused only on the robustness for system uncertainties, while seldom research has dealt with the response speed of the controlled systems. In view of the above literature review and the challenging problem we discussed, this paper proposes an innovative nonlinear control technique, named fractional order signed power feedback control, that achieves both robustness and fast response. The nonlinear control with signed power function, which is widely used in sliding-mode control, with fast response dynamic, and the framework of Bode's ideal transfer function with robust damping ratio are taken as the ideological basis of our control approach. In this paper we propose an innovative feedback control strategy that combines linear fractional order robust control as well as nonlinear feedback control in an effective way to exploit both of their advantages. We call this nonlinear control strategy the fractional order signed power feedback law. On the other hand, although the nonlinear fractional order control is not difficult to be realized in figuration and simulation, it should be noticed that analysis of nonlinear FOSs remains extremely difficult due to the lack of any developed system theory and any efficient mathematical tool, and research on this topic is insufficient yet. For example, although the Lyapunov direct method for fractional order nonlinear dynamic systems [19] has already been proposed, fractional derivative is required for the compound Lyapunov function, which is rather complicated in most situations. Furthermore, proper Lyapunov function is still difficult to be found and the related analysis is complicated even for those integer order systems with nonlinear power feedback laws. Meanwhile, it is not an easy task to quantitatively analyze the overshoot and rise time for nonlinear systems, and seldom literatures are reported so far. Facing the great difficulties in nonlinear FOSs analysis, the comparison theorems and rise time estimation method for a class of nonlinear FOSs are proposed in this paper. Then the fractional order nonlinear system rise time can be theoretically analyzed utilizing those theorems. The rest of this paper is arranged as follows. The next section introduces some basic definitions and the problem formulation we discussed in this paper, as well as some preliminaries that will be used in the later sections. Section 3 proposes the fractional order signed power feedback laws for both tracking and regulation problems. In order to analytically analyze the time response of such nonlinear FOS, two comparison theorems as well as rise time estimation theorems are proposed. By the table look-up, one can allocate the controller parameters to achieve desired robustness and response speed. The numerical examples are given in Section 4 to illustrate the effectiveness of the controller design method and the comparison and estimation method proposed in this paper. The main contributions and conclusions are conducted in the final section.

2. Problem formulation and preliminaries

Consider a single input single output (SISO) linear system with transfer function G( s ), the minimum phase system {A , B , C} with C ¼1 and B as a non-zero scalar in the range of this paper is expressed by fractional differential equations as

α

1 Γ( m − α)

∫0

f ( m) ( τ )

t

( t − τ )α − m + 1

dτ , (2)

where m − 1 < α < m , m ∈  . u˜ ( t ) is the nonlinear feedback control law for G( s ). Generally, system states are variables that could uniquely determine the system information at arbitrary future time along with the system input. However, as FOS is essentially an infinite dimension system [20], incompleteness problem occurs when one tries to describe the FOS by the existing definitions. That is, the finite given system states in FOS cannot reflect the whole system's information [21]. Thus, compared to the complete states in finite dimension integer order systems, those incomplete states in FOSs are called ‘pseudo states’. Nevertheless, those pseudo states can also be part of real states in FOSs. In model (1), x( t ), y( t ) are the pseudo states and output of the system, respectively. In this situation x( t ) is also the real state that represents the system output.

The real fractional order α is ranged in ( 1, 2). In this paper the system overshoot P.O. is defined as the first percentage overshoot of the step response in tracking or in regulation,

P . O. =

Mp − yv yv − y( 0)

× 100%,

(3)

where Mp , yv and y( 0) represent the output peak value, final value and the initial value respectively. The system output rise time tr for the tracking problem is taken by y(t) to change from 0 to 100p% of its final value, where p is a prespecified constant arranged in ( 0, 1⎤⎦.

⎫ ⎧ tr = min⎨ arg min y( t ) − 100p%⎡⎣ yv − y( 0)⎤⎦ ⎬ . ⎭ ⎩ t>0

(4)

Similarly, the fall time tf is defined for regulation problem as

⎧ ⎫ t f = min⎨ arg min y( t ) − 100q%y( 0) ⎬ , ⎩ t>0 ⎭

(5)

where q is constant arranged in ⎡⎣ 0, 1). Without loss of generality, those two constants are specified as p¼ 0.9 and q ¼0.1 in this paper by default. Obviously in this minimum phase system there holds x( t ) = y( t ). Hence, state x( t ) represents the output y( t ) in the rest of this paper by default. Our task is to design a nonlinear fractional order controller u˜ ( t ) for linear system G( s ) with parameter disturbance, so that the closed-loop system (1) satisfies a constant overshoot as well as a rapid rise time. 2.2. Preliminaries As a robust control strategy, CRONE provides a fractional order approach for the robust control of uncertain plants under common unity-feedback configuration (Fig. 1) [15], where r ( t ) is the reference input of the system, du( t ), dy( t ) and dm( t ) are external

2.1. Problem formulation

α ⎧ ˜ ( t ), ⎪ D x( t ) = Ax( t ) + Bu ⎨ ⎪ ⎩ y( t ) = x( t ),

Dαf ( t ) =

(1)

where D is in the sense of Caputo's fractional order derivative definition as

disturbances, and y( t ) is the output of the closed-loop system. For simplicity, the designing principle of second generation CRONE approach is to seek the synthesis of such an open-loop transfer function temple

F ( s ) = C ( s )G( s ) =

⎛ ωcg ⎞α K ⎜ ⎟ ≜ , ⎝ s ⎠ sα

(6)

where real fractional order α ∈ ( 1, 2) and positive system gain K > 0. It is in fact Bode's ideal open loop transfer function, and indicates the vertical template in an open-loop Nichols locus. This vertical displacement of the template ensures the robustness of phase margin φm = ( 2 − α )π /2, which conveys the stability degree

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Fig. 2. Nonlinear fractional order control block diagram.

Fig. 1. General CRONE control block diagram.

robustness [15]. In fact for those disturbed gains, the phases keep unchange while amplitudes are parallel lines with slope −20αdB/dec nearby ωcg, which ensure unchanged phase margin and good robustness to those disturbances. What is more, the form and the vertical sliding of the template also ensure the constancy of the resonant peak Mr = 1/sin( απ /2), which indicates the constancy of reduced first overshoot of the step response in tracking or in regulation, through the tangency of the template to the same iso-overshoot contour, a performance contour which is a Nichols amplitude contour that have validated as an iso-overshoot contour by using complex noninteger integration [22]. Besides, the constancy of the damping ratio ζ = − cos( π /α ) in tracking and in regulation can also be guaranteed. Then the fractional order controller C ( s ) can be obtained from (6) as

C ( s ) = F ( s )G−1( s ).

(7)

Meanwhile, the Mittag–Leffler function in two parameters defined as follows is frequently used in the analysis of FOSs ∞

Eα, β( z ) =

∑ k=0

zk Γ ( β + αk )

(8)

with α , β , z ∈  and Re( α ) , Re( β ) > 0. Especially, if β = 1, the two parameter Mittag–Leffler function degenerates to classical one parameter Mittag–Leffler function, expressed as Eα ( z ) = Eα,1( z ). Lemma 1 ([23]). SISO FOS D αx( t ) = Ax( t ) + Bu( t ) with uniformly continuous step input u( t ) = R ∈  , t ≥ 0 has unique solution

x( t ) = Eα( At α )x( 0) + RBt αEα, α + 1( At α )

(9)

closed-loop system. That is to say, this fractional order controller can be easily obtained from (7), and the matrices A , B and C for minimal phase system are

A = 0,

u˜ ( t ) = e( t )

holds as well. Lemma 2 ([24,25]). Mittag–Leffler function of negative argument Eα ( −x ) is completely monotonic for all 0 ≤ α ≤ 1, and the generalized Mittag–Leffler function Eα, β ( −x ) with real variable x possesses the complete monotonicity property for 0 ≤ α ≤ 1, β ≥ α .

3. Main results 3.1. Fractional nonlinear controller design In order to obtain a robust and fast control system for the linear plant G( s ) with gain disturbances, a linear fractional order controller C ( s ) and a nonlinear feedback control unit · β sgn( ·) are introduced, as shown in Fig. 2. β is independent from the fractional order α that can be treated as a nonlinear coefficient, which is relevant to the response speed. Among them, the linear controller C ( s ) is designed as a CRONE controller in order to satisfy the robustness demand, represented as a constant overshoot, for the

β

r( t )

1− β

(

) (

)

sgn e( t ) sgn r ( t ) ,

(12)

where r ( t ) is the reference input, e( t ) = r ( t ) − y( t ) is the system error and β ∈ ( 0, 1) is an independent real nonlinear coefficient. Case 2: r ( t ) = 0 and x( 0) ≠ 0

u˜ ( t ) = − x( 0)

x( t )

β

x( 0)

(

)

sgn x( t ) ,

(13)

where nonlinear coefficient is in the range β ∈ ( 0, 1). Along with the linear controller C ( s ), this control strategy is named the fractional order signed power feedback law. By so designing, the closed-loop system with fractional order signed power feedback law could achieve a constant overshoot as well as a fast rise time under the plant's gain disturbance. In fact, for all r ( t ) ≠ 0,

e( t ) r( t )

=

r ( t ) − y( t ) r( t )

≤1 (14)

always holds when the overshoot is less than 100%, and therefore ensures the inequation

r( t ) (10)

(11)

C = 1,

through the pseudo state space realization from transfer function. The fractional order α is chosen to certify the demanded robustness. On the other hand, the nonlinear feedback control law u˜ ( t ) is defined as: Case 1: r ( t ) ≠ 0, x( 0) = 0

under initial condition x( 0). Moreover, if A ≠ 0, another expression

⎡ RB ⎤ RB α x( t ) = ⎢ x( 0) + ⎥Eα( At ) − ⎣ A⎦ A

B = K,

e( t ) r( t )

β

≥ r( t )

e( t ) r( t )

= e( t )

(15)

for all β ∈ ( 0, 1). The difference between the linear feedback approach and the fractional order signed power feedback law proposed in this paper is the embedding of the nonlinear process · β sgn( ·), which is represented as the signed power control law u˜ ( t ). It can be seen that the fractional order signed power feedback law is in fact a nonlinear amplification for the linear feedback error e( t ). Then the nonlinear coefficient β can be regarded as a parameter relevant to the amplification factor. In linear feedback situation, the controlled quantity output is linearly proportional to the system error e( t ). A large error leads to a large controlled quantity. However, when introducing the nonlinear feedback strategy, the input error signal e( t ) for the controller is factitiously amplified. Thus, with the same amount of e( t ), it could lead to an amplified controlled quantity for nonlinear feedback than the linear one. Therefore the response speed is promoted. The amplification is obvious when system error e( t ) is large and it will become unconspicuous when e( t ) is small so that the system will not oscillate heavily. To the contrary, if the nonlinear feedback is a shrink for the error, the controlled quantity will be shrunk and therefore the response speed will be

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slowed down. Indeed, the robustness of the general linear control system has already been studied in literatures. The loop will be robust to plant gain variations, since even though the gain crossover frequency may change, the plant phase margin will not, and neither will the controller phase. That devise a controller that ensures a constant open-loop phase. Despite the nonlinear feedback, the open-loop system is a standard Bode's ideal transfer function and thus the robustness of phase and gain margin, as well as the damping ratio and resonant peak, keeps unchange. From the description (12) of the signed power control law, it can be seen that nonlinear feedback u˜ ( t ) has no influence on the system's phase and no significant influence on the gain in frequency domain. The robustness of this nonlinear system can be guaranteed. Then the resonant peak Mr defined by the maximal value of the complementary sensitivity function T ( s ) = F ( s ) /⎡⎣ 1 + F ( s )⎤⎦, where F ( s ) is the open-loop transfer function, and indicates the first percent overshoot in tracking and regulation [26], will be discussed. Obviously, the resonant peak is only depending on the fractional order α, while the other parameters as well as system disturbances have no effect on it. That ensures an almost constant overshoot despite of the disturbances. When the nonlinear feedback is introduced, it should be noticed that the open-loop system keeps unchange and the amplification of the error will not markedly affect the complementary sensitivity function T ( s ). Thus, the nonlinear feedback has little influence on the resonant peak and the first overshoot. The closed-loop system also ensures a good robustness to the parameter disturbances. It can be seen that when considering the external disturbances as shown in Fig. 1, those disturbances du( t ), dy( t ) and dm( t ) can all be regarded as the system gain disturbance if they are multiplicative perturbations. Thus, the control scheme for those system gain disturbances can also be used for those external disturbances. Take the load disturbance rejection problem as an example. The external load disturbance is a classical gain disturbance in control systems while the linear controller C ( s ) ensures nearly consistent dynamic characteristics in time response. As a consequence, this kind of disturbances can always been rejected by the linear controller C ( s ). Remark 1. This controller shares some similarities with high-gain adaptive controller [27]. The adaptive gain in high-gain adaptive control, or the signed power function in our method, is high when error signal is close to zero. However, the signed power function is not a non-decreasing function of time t. In fact this fractional order signed power feedback law always ensures a moderate controlled quantity under whether large or small system error. 3.2. Comparison theorems for a class of fractional differential equations Before analyzing the system rise time under fractional order signed power feedback law, two comparison theorems for a class of fractional order systems are introduced as follows. Theorem 1. Linear functions

f1( x ), f2 ( x ) and two time-varied functions x( t ), y( t ) satisfy the fractional differential equations

⎧ D x( t ) = f ( x) = 1 − k x( t ) 1 1 ⎨ α ⎪ ⎩ D y( t ) = f2 ( y) = 1 − k2y( t ) ⎪

α

Case 2: 1 < α < 2. Firstly, the solutions of x( t ) and y( t ) can be obtained easily from Lemma 1 and satisfy x( t ) > 0, y( t ) > 0. Then a real function

z( t ) = x( t ) − y( t ) can be defined. As defined in (16), inequation f1( x ) ≥ f2 ( x ) for positive x indicates the coefficients relationship

k2 ≥ k1. Then by introducing the Laplace transform, there is

( ) = k2( Y ( s ) − X ( s )) + ( k2 − k1)X ( s ).

s αZ ( s ) = s α X ( s ) − Y ( s ) = k2Y ( s ) − k1X ( s ) (17)

Therefore,

Z ( s) =

( k2 − k1)X ( s) . s α + k2

(18)

Indeed, by adding an integral segment and taking the Laplace inverse transform, there holds equations

Z ( s) ( k2 − k1)X ( s) = k − k X ( s)Y ( s), = ( 2 1) s s( s α + k 2 )

∫0

t

(19)

z( τ )dτ = ( k2 − k1)x( t )*y( t ) > 0,

(20)

where n is the convolution operation. Obviously for the initial stage (when t is small),

t

∫0 z ( τ )dτ > 0

indicates z( t ) ≥ 0. Notice from the analytical solution of x( t ) and

y( t ) that both x( t ) and y( t ) are monotone increasing functions before reaching the peak value. Therefore z( t ) have to be a nonnegative function in this region, as

( k2 − k1)x( t )⁎y( t ) is a positive

increasing function. Thus, z( t ) ≥ 0 holds in that region and it indicates inequation x( t ) ≥ y( t ).

□

Remark 2. The conclusion still holds if all the “ ≥” symbols are changed to “ ≤” in Theorem 1. The relevant proof is omitted here since it is similar to the former one. Theorem 2. Consider two fractional differential equations with x( 0) = x′( 0) = y( 0) = y′( 0) = 0

⎧ Dαx t = f x t , () () ⎪ ⎨ α ⎪ ⎩ D y( t ) = g y( t ) = c − ky( t ),

( (

) )

where k > 0 and α ∈ ( 0, 2), x( t ) and y( t ) are continuous in t for all t ≥ 0. Let T ≜ ⎣⎡ 0, tm⎤⎦ (tm could be infinity) be the maximal interval of existence of the solution x( t ) and y( t ). And suppose x( t ), y( t ) ∈ Ω for all t ∈ T . If y( t ) satisfies the differential inequality

(

)

D α y( t ) ≥ f y( t )

with y( t ) ∈ Ω for all t ∈ T . Then, y( t ) ≥ x( t ) holds for all t ∈ Λ, where

Λ = T ∩ T0,

(21)

T0 = ⎡⎣ 0, t0⎤⎦,

(22)

⎧ ⎫ t0 = min⎨ arg min N ( t , k ) ⎬ , ⎩ t≥0 ⎭

(23)

(16)

α ∈ ( 0, 2), k1, k2 > 0 and zero initial conditions x( 0) = x′( 0) = y( 0) = y′( 0) = 0. Then x( t ) ≥ y( t ) holds before x( t ), y( t ) reaching their first peak values if f1( x ) ≥ f2 ( x ).

with

Proof. Case 1: 0 < α ≤ 1. From Lemma 1 and 2, the conclusion can be proved easily when 0 < α ≤ 1 as the Mittag–Leffler function is completely monotonous.

where function argminxf ( x ) represents the value of variable x when function f ( x ) reaches its minimal value. For those non-one-to-one mapping

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function f ( x ), min{argminxf ( x )} is the minimal x among all the x values that lead to a minimal f ( x ) value. The function N ( t , k ) is defined as

feedback law proposed in Section 3.1 can be analyzed and estimated by several theorems below.

N ( t , k ) = Eα, α( −kt α ).

Theorem 3. In the presence of unit step response with zero initial conditions, the rise time tr of system (1) under fractional order signed power feedback law (11) and (12) can be estimated as

(24)

Proof. Define a non-negative function m( t ) ≥ 0 in T that satisfies f x ( t ) = g x ( t ) − m( t ) . Then one has

(

)

(

)

Dα x( t ) = c − kx( t ) − m( t ).

(25)

By introducing the Laplace transform, the equation becomes n− 1 α

s X ( s) −

∑s

c x ( k)( 0) = s αX ( s ) = − kX ( s ) − M ( s ), s

α−k−1

k=0

(26)

where X ( s ), Y ( s ) and M ( s ) are the Laplace transforms of x( t ), y( t ) and m( t ), respectively and integer n satisfies n − 1 ≤ α < n. Meanwhile for function y( t ) there holds n− 1

s αY ( s ) −

∑ s α − k − 1y( k) ( 0) = s αY ( s) =

k=0

(31)

if tr , tr0 ∈ Λ, where Λ is defined in (21). Furthermore, rise time tr is defined in (4), while tr0 has the representation as

⎧ ⎫ tr 0 = min⎨ arg min Kt αEα, α + 1( −Kt α ) − p ⎬ . ⎩ t>0 ⎭

(27)

s α( Y ( s ) − X ( s )) = − k⎡⎣ Y ( s ) − X ( s )⎤⎦ + M ( s ).

(28)

Hereafter, another function z( t ) = y( t ) − x( t ) can be defined. The initial condition and the Laplace transform for this function are obvious as z( 0) = y( 0) − x( 0) = 0, z′( 0) = y′( 0) − x′( 0) = 0 and Z ( s ) = Y ( s ) − X ( s ). Thus,

M( s) . sα + k

(29)

By applying the inverse Laplace transformation to (29), we can get

z( t ) = ⎡⎣ t α − 1Eα, α( −kt α )⎤⎦⁎m( t ) = N ( t , k )⁎m( t ).

Proof. The closed-loop system with fractional order signed power feedback law can be expressed as β ⎧ α ⎪ D x( t ) = K e( t ) sgn e( t ) , ⎨ ⎪ ⎩ y( t ) = x( t ),

(34)

β Dα x( t ) = K ⎡⎣ 1 − x( t )⎤⎦ ,

(35)

as well as two linear functions

Obviously z( t ) = y( t ) − x( t ) ≥ 0 can be guaranteed as long as

f ( x) = K ( 1 − x) .

N ( t , k ) with constant gain k is a uniformly continuous function for t that satisfies N ( 0, k ) > 0. Thus, there exists a range T0 = ⎡⎣ 0, t0⎤⎦ that ensures N ( t , k ) ≥ 0, where

Therefore,

α ∈ ( 0, 2).

y( t ) ≥ x( t ) is guaranteed in □

Λ = T ∩ T0 for all

Remark 3. If the inequation in Theorem 2 is changed into

)

β

(37)

Indeed, there are

(

α

)

D x 0( t ) = f0 x 0( t ) ,

(

)

Dα x( t ) = f x( t ) .

(38) (39)

Obviously, inequation

(40)

holds in region x ∈ Ω ≜ ⎡⎣ 0, 100p%⎤⎦. By utilizing Theorem 2, it is easy to get

It is noticed is that if T ⊂ T0 , or to say tm ≤ t0 , then Λ = T .

(

(36)

f0 ( x) ≤ f ( x)

⎧ ⎫ t0 = min⎨ arg min N ( t , k ) ⎬ . ⎩ t≥0 ⎭

(33)

Dα x 0( t ) = K ⎡⎣ 1 − x 0( t )⎤⎦,

f0 ( x) = K (1 − x),

α ∈ ( 1, 2). According to the definition, Mittag–Leffler function

)

where e( t ) = r ( t ) − y( t ). Take the time period before the output y( t ), or to say the pseudo state x( t ), reaching p ¼0.9 into consideration. Then let us introduce two fractional differential equations under zero initial conditions

(30)

N ( t , k ) ≥ 0. Indeed, when α ∈ ( 0, 1⎤⎦, N ( t , k ) ≥ 0 always established as the monotonicity property of Mittag–Leffler function is guaranteed in Lemma 2. Then we only need to consider those

(32)

p is acquiescently specified as 0.9 in the definition (4).

(

c − kY ( s ) s

in Laplace domain with n − 1 ≤ α < n. Then subtract (26) from (27), there is

Z ( s) =

tr ≤ tr0,

(

)

D αy( t ) ≤ f y( t ) , or to say D αx( t ) ≥ g x( t ) , the conclusion turns to y( t ) ≤ x( t ) without other changes. The proof is similar. Remark 4. From the process of the proof it can be found that t ∈ Λ is only a sufficient but not necessary condition for the conclusion, as N ( t , k ) < 0 does not represent z( t ) < 0. That is where the conservatism comes from. 3.3. Rise time estimation method Based on the analysis in Section 3.2, output rise time for closedloop system {A , B , u˜ ( t )} under fractional order signed power

x 0( t ) ≤ x( t )

(41)

in t ∈ Λ . In this situation region Λ = T ∩ ⎡⎣ 0, t0⎤⎦ and t0 is defined in (23) with k ¼K. Apparently, tr is the rise time of x( t ) with x( 0) = 0. Meanwhile, tr0 can be defined as rise time of x0( t ) with x0( 0) = 0 under definition (4) as well. Therefore tr0 has the representation as (32). As suggested in the theorem, there holds tr , tr0 ∈ Λ. It is obvious from (41) that p = 0.9 = x0( tr 0) ≤ x( tr 0) and x( tr ) = p = 0.9. That is,

p = 0.9 = x( tr ) ≤ x( tr0). It is known that x( t ) is monotonic increasing function in □ there holds tr ≤ tr0 .

(42)

Λ. Thus,

Since the inverse Mittag–Leffler function is difficult to be analytically expressed, furthermore, both the N ( t , k ) in Theorem 2 and

x( t ) in Lemma 1 are non-monotonic functions as α ∈ ( 1, 2), it is

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6

Table 1 Evaluation of t0 ( s ) with k ¼K.

k1 = β,

α

1.1

1.3

1.5

1.7

1.9

K¼ 1 K¼ 10 K¼ 100

4.3980 0.5422 0.0669

3.1900 0.5427 0.0923

2.9530 0.6362 0.1371

2.9580 0.7636 0.1971

3.0660 0.9125 0.2716

k2 =

Remark 5. tr0 is in fact rise time for traditional CRONE approach (6). This theorem indicates that the closed-loop system with the fractional order signed power feedback law proposed in this paper has a faster response, represented by the rise time, than the CRONE approach. Theorem 4. Consider the zero-input responses for system (1) under fractional signed power feedback law (11) and (13) with x( 0) ≠ 0. The fall time t f is defined in (5). Then t f can be estimated as

tf ≤ tf 0 if t f , t f 0 ∈ Λ, where

(43)

Λ is defined in (21), and t f 0 is defined as

⎧ ⎫ t f 0 = min⎨ arg min Eα( −Kt α ) − q ⎬ . ⎩ t>0 ⎭ The value of q is specified in definition (5) as 0.1.

In fact the fractional order nonlinear closed-loop system in this theorem can be expressed as

(

) (

)

(45)

Then the proof of this theorem is not described further as it is similar to Theorem 3. What should be noticed is that t f 0 is in fact the fall time of the traditional CRONE approach under the same parameters. Like Remark 5, a faster response speed can be proved in regulation problems then. Although the fastness of response speed has been proved above, a more accurate rise time estimation method is much more meaningful in engineering practices, which could guide engineers to optimal the controller parameters. There are two theorem that could help us to estimate the system rise time. Theorem 5. Rise time tr of system (1) with unit step reference input and fractional order signed power feedback law (11) and (12) under zero initial condition can be estimated as

tr1 ≤ tr ≤ tr2,

(46)

where

⎧ ⎫ tri = min⎨ arg min Bit αEα, α + 1( Ai t α ) − p ⎬ , ⎩ t>0 ⎭

i = 1, 2,

(47)

and

⎡ A B ⎤ ⎡ −k K K ⎤ ⎥, ⎢ 1 1⎥ = ⎢ 1 ⎣ A2 B2 ⎦ ⎣⎢ −k2K K ⎥⎦

p

.

(49)

Dα x1( t ) = K ( 1 − k1x1), x′1( 0) = 0,

(50)

Dα x2( t ) = K ( 1 − k2x2), x2( 0) = 0,

x′2 ( 0) = 0.

(51)

Similarly, like the proof of Theorem 3, two linear functions f1( x ) and f2 ( x ) are introduced so that (50) and (51) can be represented as

(

)

(52)

(

)

(53)

Dα x1( t ) = f1 x1( t ) ,

Dα x2( t ) = f2 x2( t ) , in which

fi ( x) = K ( 1 − kix),

i = 1, 2.

(54)

Now consider the nonlinear function (37) and linear functions (54). Obviously there holds

f2 ( x) ≤ f ( x) ≤ f1( x)

(55)

in region Ω = ⎡⎣ 0, 100p%⎤⎦. Then the proof is similar as Theorem 3. Inequation x2( t ) ≤ x( t ) holds in t ∈ Λ = T ∩ ⎡⎣ 0, t02⎤⎦, where

(44)

⎧ α β 1− β ⎪ D x( t ) = − K x( 0) x( t ) sgn x( 0) sgn x( t ) , ⎨ ⎪ ⎩ y( t ) = x( t ).

β

Proof. First of all, let us introduce two fractional order differential equations with zero initial conditions as follows

x1( 0) = 0, complicated to solve analytical expression of t0 and tm in the proving process. However, a tabulation can help us acquire the value. We can get the t0 value according to the following tables. It is easy to find that tm < t0 always holds under those parameters. Thus, there is Λ = T ∩ T0 = T under those constrains. Table 1 shows the evaluation of t0 under different gain K and fractional order α.

1 − ( 1 − p)

(48)

⎧ ⎫ t02 = min⎨ arg N ( t , k2K ) ⎬ . ⎩ t≥0 ⎭

(56)

Similarly, the value of t02 can be estimated under different α and β values as shown in Table 2. It is easy to find that there is T ∩ ⎡⎣ 0, t02⎤⎦ = T in this situation. The same goes for the proof of x( t ) ≤ x1( t ). In conclusion, we can get

x2( t ) ≤ x( t ) ≤ x1( t )

(57)

Table 2 Evaluation of t02 ( s ) with k = k2K . β

α 1.1

1.3

1.5

1.7

1.9

6.3330 4.3470 3.6750 3.3590

5.3500 3.8610 3.3380 3.0880

4.9980 3.7480 3.2960 3.0780

4.9010 3.7890 3.3770 3.1760

1.0775 0.7395 0.6250 0.5715

1.1525 0.8320 0.7190 0.6650

1.2900 0.9670 0.8505 0.7940

1.4585 1.1275 1.0050 0.9450

0.1833 0.1258 0.1063 0.0972

0.2483 0.1792 0.1549 0.1433

0.3329 0.2496 0.2195 0.2050

0.4342 0.3356 0.2992 0.2813

Case 1: K¼1 0.2 0.4 0.6 0.8

9.8910 6.3400 5.1980 4.6750

Case 2: K¼10 0.2 0.4 0.6 0.8

1.2190 0.7815 0.6405 0.5760

Case 3: K¼100 0.2 0.4 0.6 0.8

0.1503 0.0963 0.0790 0.0710

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in region Ω. Define tr1 and tr2 as the rise time of fractional differential equations (52) and (53). Then obviously there holds tr1 ≤ tr ≤ tr 2. What is more, the analytical expression of tr1 and tr2 can be obtained by Lemma 1. The solutions of (52) and (53) are

xi ( t ) = Bit αEα, α + 1( Ai t α ).

(58)

with i¼1,2. Coefficients Ai and Bi are expressed as Eqs. (52) and (53). Therefore tri satisfies the equation

⎫ ⎧ tri = min⎨ argmin Bit αEα, α + 1( Ai t α ) − p ⎬ . ⎭ ⎩ t>0

(59)

sign of r ( t ) has already been considered in the design of the feedback control law. Theorem 6. Fall time t f of the closed-loop system (1) with fractional order signed power feedback law (11) and (13) can be estimated as

(60)

  

)

m margin φm , the fractional order can be obtained as α = . π α The nominal open-loop gain K can be obtained as K = ωcg , where ωcg is the desired crossover frequency in practices. The linear controller C ( s ) has the representation as (7), while parameters in F ( s ) are determined by the above steps, and G( s ) is the nominal transfer function of the plant. By checking the table in the Appendix with fixed α and K value, the nonlinear feedback coefficient β can be obtained with specification on the system rise time. Thus, all the parameters needed in the robust fast controller design is obtained.

Besides, there are some specific statements for the selection of parameters α and β that we should notice. (i) The closed-loop system (1) will become a half open loopsystem if we choose β = 0 in the nonlinear feedback control law. In that situation the pseudo state space equation of the

(

)

closed-loop system becomes D αx( t ) = K sgn r ( t ) − x( t ) = ± K . No matter what value the symbol function is, the system is unstable. As a result, chattering phenomenon will happen in

where

⎧ ⎫ t fi = min⎨ arg min Fi( t ) ⎬ , ⎩ t>0 ⎭

i = 1, 2,

Fi( t ) = Eα( Ai t α )x( 0) + Bit αEα, α + 1( Ai t α ) − qx( 0),

(61)

(62)

and

⎡ A B ⎤ ⎡ −k K c K ⎤ 1 ⎥, ⎢ 1 1⎥ = ⎢ 1 ⎣ A2 B2 ⎦ ⎣⎢ −k2K c2K ⎥⎦

(63)

c1 = β − 1, β

k2 =

phase margin specifications. That is, for the expected phase

(

It should be noticed that r ( t ) is defined as the unit step input by default in Theorems 1 and 2. However, the conclusions still hold without the restriction on r ( t ). That is because the amplitude and

k1 = β,

 Determine the fractional order α based on the overshoot, or the 2 π−φ

Remark 7. In fact inequation x0( t ) ≤ x2( t ) ≤ x( t ) ≤ x1( t ) can be proposed by utilizing Theorem 1. As a consequence, there is

t f 1 ≤ t f ≤ t f 2,

comparison, tr0 is also given as the rise time for CRONE control approach under same parameters. One can collocate the independent controller parameters α and β based on the demands for robustness and response speed. Here is a controller parameter design procedure for those linear systems with gain uncertainties.

□

Remark 6. From the proof above we find that the restriction of t ∈ T ∩ T0 can be ignored under the designed parameters in this paper.

tr1 ≤ tr ≤ tr2 ≤ tr 0.

7

1−q , 1−q

β

c2 =

q−q . 1−q

(64)

The proof is similar to Theorem 5 and is not described in this paper. Remark 8. What can be further obtained from the two theorems above is that a smaller β value indicates a faster response speed, or to say a smaller rise time tr , or the fall time t f accordingly. As β is independent from the other parameters such as system gain k and fractional order α, one can simply obtain a faster response speed by choosing a smaller β value. Perceptually speaking from inequation (15), a smaller β represents a larger amplification of the system error e( t ) and a larger controlled quantity. Consequently, it could lead to a faster response speed in transient response. Although the analytic solution for nonlinear fractional differential equations are too difficult to be obtained, it can be estimated and approximated by two linear fractional differential equations utilizing Theorem 4 or 6. Further, consider the rise time of the control system proposed in this paper, as well as the estimated rise time, is also difficult to be analytically expressed, a tabulation look-up is given in the Appendix, while the accurate value of tr can been obtained by SIMULINK in MATLAB. By

that case. Similarly that might happen when β is close enough to zero. That is what we should avoid in practice. (ii) The fractional order signed power feedback control law proposed in Section 3.1 will degenerate into traditional linear controller if β = 1. (iii) The controlled system will become integral order one if the fractional order α is chosen as 1 or 2.

4. Numerical examples Firstly, some simulations for the closed-loop system (1) with different particular K and β values are given to provide visual representations of the step response in Fig. 3. As a comparison, the step responses for CRONE approach are also given. From the figure we can see that the advantage in rise time for the system under fractional order signed power feedback control law is obvious. Moreover, by checking the table in the Appendix we could find that the estimation of rise time is fairly accurate. Take K = 10, α = 1.5, β = 0.6 as an example, the closed-loop system's rise time is estimated as ⎡⎣ 0.2758 s, 0.2954 s⎤⎦ while the exact value is 0.2830 s. The error is about 6.9% in that situation. Compared with the rise time tr0 = 0.3141 s for linear control system, the advantage is obvious. On the other hand, some corollaries can be obtained from the Appendix. It can be seen that the estimation value becomes more accurate as α value increases. Meanwhile, although a larger β leads to a larger rise time tr , the relationship between α and tr is not positive correlated. 4.1. Tracking problem Consider the quarter-car model in Fig. 4 from [28]. m2 is the mass supported by each wheel, and is taken as equal to a quarter of the total mass of the body. k1 is the stiffness of the spring, b1 is the damping coefficient. m1 is the unsprung mass, z0( t ) is the

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Fig. 3. Unit step response of closed-loop system under fractional order signed power feedback law and linear CRONE approach with order α = 1.5.

H ( s) =

Z2( s ) C ( s) = Z1( s ) m2s 2 + C ( s )

(66)

with the plant

G( s ) =

(67)

where 150 kg ≤ m2 ≤ 300 kg is the uncertain mass. It is a second order system and the output might oscillate without any controller. In order to robustly control the displacement of mass m2, which means to get a constant displacement overshoot despite the variation of m2, the linear controller C ( s ) is designed as

Fig. 4. Two degree-of-freedom model.

C ( s) = deflexion of the road, z1( t ) and z2( t ) are the vertical displacements of the wheel and body respectively. The article had developed the force T ( s ) on the mechanical system as

T ( s ) = C ( s )⎡⎣ Z1( s ) − Z2( s )⎤⎦,

1 , m2s 2

(65)

where C ( s ) is the controller. To analyze the vibration insulation of the sprung mass, the author had obtained the dynamic equations of the system and the closed-loop transfer function as

⎛ ωu ⎞α −1 ⎛ ωu ⎞α ⎜ ⎟ G ( s) = ⎜ ⎟ m s 2, ⎝ s ⎠ ⎝ s ⎠ 2

where ωu =

1 1/1α

( m2ω02−α)

(68)

is the open loop unit gain frequency with

ω0 = 12.16 × 10−5 rad, and α = 1.2. Then for the nonlinear feedback control law u˜ ( t ), we had chosen the coefficient β as 0.5 in order to get a smaller rise time, which indicates a faster response. That is to say,

u˜ ( t ) = 1 − z2( t )

0.5

(

)

sgn 1 − z2( t ) .

(69)

The linear controller C ( s ) and nonlinear feedback control law u˜ ( t ) have then comprised the fractional order signed power feedback law. The step response for the closed-loop system is shown in the left in Fig. 5.

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Fig. 5. Step response of sprung mass of the system under fractional order signed power feedback law (a) and linear CRONE approach (b) with β = 0.5.

The CRONE control method is also provided as a comparison. Related analysis can be found in [28]. Accordingly, step response for this linear CRONE approach is shown in the right in Fig. 5. From the figure, the advantage of rims time is obvious. Take m2 = 150 kg as an example, the rise time of the closed-loop system under fractional order signed power feedback law is less than 0.20 s while that of the linear control system is about 0.26 s. That is almost a 23% promotion for the rise time. Meanwhile, it can be seen that the overshoot of the closed-loop system remains constant despite of the mass variation, which is about 4.24%. It also shows a good performance in robustness in the time domain. Thus, the brief example illustrates the effectiveness of our controller design method. 4.2. Regulation problem Regulation problem can be seen as a kind of internal disturbance rejection problem, while the initial output value is treated as the internal disturbance. In this section the studied plant is described by the transfer function

⎛ s ⎞ ⎜1 + ⎟ ω′1 ⎠ ⎝ G( s ) = C0 ⎛ s ⎞⎛ s s2 ⎞ + 2 ⎟⎟ ⎜1 + ⎟⎜⎜ 1 + 2ς ω1 ⎠⎝ ωn ⎝ ωn ⎠

(70)

in [22]. And its nominal parametric state is defined as C0norm = 50, ω′1norm = 0.01, ω1norm = 0.02, ωnnorm = 1 and ςnorm = 0.8. The parameter C0 is uncertain within the interval C0norm/2 ≤ C0 ≤ 2C0norm . Regardless of the design of proportional integrator and lowpass filter, the controller C ( s ) is defined as

C ( s) =

β( s) , G0( s )

(71)

where

β( s) =

⎛ ωu ⎞α ⎜ ⎟ , ⎝ s ⎠

ωu = 20,

α = 1.44,

and G0( s ) denotes the nominal transfer function of the plant. Consider the closed-loop response to a step state disturbance with different C0 values under initial state x( 0) = 1, the nonlinear feedback law u˜ ( t ) is defined as (13) with β = 0.5. That means the nonlinear feedback law is chosen as β

(

)

u˜ ( t ) = − x( t ) sgn x( t ) .

(72)

Then by introducing the fractional order signed power feedback control law as above, the system state response, comparing with the CRONE approach, can be seen in Fig. 6. It is easy to find that the fall time of the nonlinear CRONE control system, about 0.063 s for C0 = C0norm , is obvious shorter than those

Fig. 6. State response of the system under fractional order signed power feedback law (a) and linear CRONE approach (b) with β = 0.5.

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of the linear CRONE control systems, about 0.074 s. Meanwhile, the robustness is illustrated by the desensitization to the parametric state of the plant, of the constant overshoot of the response.

Table 3 Rise time estimation table. β

α 1.3 Estimation

5. Conclusions

1.5

1.7

tr

Estimation

tr

Estimation

tr

1.1220

[1.1696 1.2232] [1.2200 1.3052] [1.2808 1.3712] [1.3568 1.4216]

1.1860

[1.2456 1.2844] [1.2820 1.3408] [1.3240 1.3832] [1.3736 1.4136]

1.2550

Case 1: K¼1

In this paper, a nonlinear fractional order control scheme, named fractional order signed power feedback control law, is proposed that can robustly and fast control a series of linear systems with parameters disturbances. Firstly, the controller structures for both tracking and regulation problem are given, and the robustness for the closed-loop system under such controllers is analyzed. Then, the comparison theorems for a class of fractional differential equations are proposed, which are later used to theoretically analyze the rise time performance index of the closedloop system. Meanwhile, the rise time can be estimated with a small error by the estimation method utilizing several linear fractional differential equations. Numerical examples have shown the effectiveness of the controller in overshoot maintenance and rise time (or fall time) improvement. The proposed fractional order signed power feedback control law has good effects on both the tracking for input r ( t ) and the rejection of state disturbance. The main contributions of this work are as follows.

 A nonlinear fractional order controller is proposed to improve both





the system overshoot and rise time. The method breaks through the limitation of traditional controllers. It can be easily synthesized via parameter calculation provided with performance demand. Comparison results of a class of nonlinear fractional differential equations are given. Moreover, rise time performance index is theoretically analyzed for the closed-loop systems. This conclusion can be further used in nonlinear FOSs analysis. We also proposed a rise time estimation method and then simplify our controller synthesis procedure that replaces parameter calculations by tabulation look-up, which is favorable as an engineering approach.

It is believed that the nonlinear fractional order controller provides a new avenue for the design of those systems demand for faster response speed with a good robustness for system disturbances.

Acknowledgment The work described in this paper was fully supported by the National Natural Science Foundation of China (Nos. 61573332, 61601431), the Fundamental Research Funds for the Central Universities (No. WK2100100028), the Anhui Provincial Natural Science Foundation (No. 1708085QF141) and the General Financial Grant from the China Postdoctoral Science Foundation (No. 2016M602032).

Appendix A. Rise time for closed-loop systems with fractional order signed power feedback law See Table 3.

0.2 0.4 0.6 0.8

tr0

[1.0960 1.1708] [1.1664 1.2948] [1.2568 1.4052] [1.3800 1.4976]

1.2160 1.3220 1.4395

1.5710

1.4582

1.2480 1.3145 1.3845

1.2990 1.3430 1.3880

1.4346

Case 2: K¼10 0.2 0.4 0.6 0.8

tr0

[0.1864 0.1992] [0.1984 0.2204] [0.2138 0.2392] [0.2348 0.2548] 0.2673

0.1910

[0.2520 0.2636] 0.2070 [0.2628 0.2812] 0.2245 [0.2758 0.2954] 0.2445 [0.2922 0.3064] 0.3141

0.2550 [0.3214 0.3316] 0.2690 [0.3308 0.3460] 0.2830 [0.3416 0.3570] 0.2980 [0.3546 0.3648]

0.3240 0.3350 0.3460 0.3580

0.3703

Case 3: K¼100 0.2 0.4 0.6 0.8

tr0

[0.0316 0.0340] [0.0336 0.0376] [0.0362 0.0408] [0.0398 0.0434] 0.0455

0.0320 [0.0542 0.0568] 0.0350 [0.0566 0.0606] 0.0380 [0.0594 0.0638] 0.0410 [0.0628 0.0660] 0.0677

0.0550 [0.0828 0.0856] 0.0580 [0.0854 0.0894] 0.0600 [0.0882 0.0922] 0.0640 [0.0914 0.0942]

0.0830 0.0860 0.0890 0.0920

0.0955

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