Experimental Studies of a Fractional Order Universal Adaptive Stabilizer Shayok Mukhopadhyay

Yan Li

YangQuan Chen

Electrical and Computer Engineering Department, Utah State University Logan, Utah, 84341, USA Email: [email protected]

Institute of Applied Math, School of Mathematics and System Sciences, Shandong University, Jinan 250100, P. R. China and Electrical and Computer Engineering Department, Utah State University Logan, Utah, 84341, USA Email: [email protected]

Electrical and Computer Engineering Department, Utah State University Logan, Utah, 84341, USA Email: [email protected]

Abstract—This work performs experimental verification of the asymptotic stability of two different types of fractional scalar systems by using universal adaptive stabilization as in [1]. The types of systems verified experimentally are: (I) Fractional dynamics with integer order control strategy (II) Fractional dynamics with fractional order control strategy. The Mittag-Leffler function Eα (−λtα ), ∀α ∈ (2, 3] and λ > 0 is used as a Nussbaum function as per [1]. Extensive hardware in the loop simulation of the mathematically developed results in [1] have been included in the paper. The results of the experiment serve not only to improve the understanding of fractional order universal adaptive stabilization but also proves that the methodology works well on real world systems.

I. I NTRODUCTION Adaptive control is important as it has potential for application in complex and uncertain systems. The goal of adaptive control is to meet a predetermined control objective for a given class of systems. Universal adaptive stabilization (UAS) is a special case of adaptive control. UAS guarantees that the systems are asymptotically stable [2], [3]. Fractional calculus forms an important part of control theory. In [4], the authors study the applications of fractional calculus in control systems. The author in [5] introduces the idea of a fractional PID controller. Also the author in [6] proposes the fundamental ideas used in fractional control strategies. Motivated by the application of fractional calculus concepts, we propose the fractional order UAS. This work deals with the following: 1. The asymptotic stability of two different types of fractional scalar systems is experimentally studied by using the method of universal adaptive stabilization [1]: (I) Fractional dynamics with integer order control strategy, (II) Fractional dynamics with fractional control strategy. 2. As proved in [1] it is experimentally verified that the MittagLeffler function Eα (−λtα ) is a Nussbaum function for α ∈ (2, 3] and λ > 0. 3. Following the mathematical treatment of the fractional order

UAS, are results from an extensive amount of hardware in the loop experiments. The experimental results agree with the theory presented in [1]. This paper is organized as follows. Section II, introduces the fundamentals of fractional calculus and the Nussbaum function. Section III, deals with the mathematical stability analysis of the fractional order UAS. Section IV, proposes a new Nussbaum function, which is a special case of the Mittag-Leffler function. Section V, explains the details of the hardware in-the-loop laboratory setup used to test the fractional UAS. Section VI provides the results of an exhaustive amount of experiments. Section VII provides a summary of experimental results. Section VIII and IX, which discuss the conclusion and the future efforts respectively, form the end of this paper. Section X is the appendix which includes the lemmas, theorems and corollaries from [1] which are needed to prove the results from [1] used in this paper. [1] should be consulted for detailed proofs. II. P RELIMINARIES A. Fractional Calculus and Mittag-Leffler Function Fractional calculus has an important role in modern science [7]. In this paper, we choose the Riemann-Liouville and Caputo fractional operators as our primary mathematical tools. The uniform formula of a fractional integral with α ∈ (0, 1) is defined as Z t f (τ ) 1 −α dτ, (1) a Dt f (t) = Γ(α) a (t − τ )1−α where f (t) is an arbitrary integrable function, a Dt−α is the αth fractional integrator on [a, t], and Γ(·) denotes the Gamma function. For an arbitrary real number p, the RiemannLiouville and Caputo fractional operators are defined as p a Dt f (t)

=

d[p]+1 −([p]−p+1) [ a Dt f (t)] dt[p]+1

(2)

and C p a Dt f (t)

=

−([p]−p+1) [ a Dt

A. Fractional Dynamics with Fractional Control

d[p]+1 f (t)], dt[p]+1

(3)

respectively. Here [p] stands for the integer part of p, D and D are Riemann-Liouville and Caputo fractional operators respectively. The exponential function is frequently used in the solutions of integer order systems. Similarly, a function frequently used in the solutions of fractional order systems is the MittagLeffler function. It is defined as C

Eα (z) =

∞ X k=0

k

z , Γ(kα + 1)

(4)

where α > 0. The Mittag-Leffler function in two parameters has the following form Eα,β (z) =

∞ X k=0

zk , Γ(kα + β)

(5)

where α and β are arbitrary complex numbers. When α > 0 and β = 1,Eα (z) = Eα,1 (z). Moreover, the Laplace transform of the Mittag-Leffler function in two parameters is L{tβ−1 Eα,β (−λtα )} =

sα−β , sα + λ

1

(<(s) > |λ| α ),

(6)

where s is the variable in Laplace domain, <(s) denotes the real part of s, λ ∈ R and L{·} stands for the Laplace transform. B. Nussbaum Function Definition 2.1: A piecewise right continuous and locally Lipschitz function N (·) : [k 0 , ∞) → R is called a Nussbaum function if 1 sup k>k0 k − k0

Zk N (τ )dτ = +∞ k0

and 1 inf k>k0 k − k0

Zk

In this section we propose a real fractional system as in (7), with the control law given by (8). In reality, such a model would correspond to the case of a vehicle running on water or any agent moving about in a complex environment. Consider the fractional dynamics ( x(t) ˙ = a1 x(t) + a2 0 Dtα x(t) + bu(t) (7) y(t) = cx(t), x(0) = x0 with the fractional control strategy ( u(t) = −N (k(t))y(t) − (cb)−1 a2 0 Dtα y(t) ˙ k(t) = y 2 (t), k(0) ∈ R (8) where a1 , a2 , b, c, x0 ∈ R are unknown, N (k) is an arbitrary Nussbaum function with respect to k, and α ∈ (0, 1). The effect of varying a2 , b and c on the performance of the UAS can be seen in section VI. Note that if a2 = 0 then the system in (7) reduces to an integer order system and the corresponding integer order control law would be obtained by substituting a2 = 0 in (8). Applying (8) to (7) yields the closed-loop system ( y(t) ˙ = [a1 − cbN (k(t))]y(t) (9) ˙k(t) = y 2 (t), k(0) ∈ R. By using Lemma 3.1 in [1], it is obvious that k(t) is bounded and y(t) converges to zero as t → ∞. IV. N USSBAUM F UNCTION WITH M ITTAG -L EFFLER F ORM The Nussbaum function plays a crucial role in UAS [2], [8], [9], [10], [11], [12], [13]. This section shows that the Mittag-Leffler function Eα (−λtα ) for λ > 0 and α ∈ (2, 3] is a Nussbaum function, following results are as in [1]. Lemma 4.1: Suppose that f (k) is bounded for all k and N (k) is a Nussbaum function with respect to k. Then f (k) + N (k) is also a Nussbaum function with respect to k. Lemma 4.2: ek cos(k) is a Nussbaum function with respect to k. Theorem 4.1: Eα (−λk α ) is a Nussbaum function with respect to k, where α ∈ (2, 3] and λ > 0. V. E XPERIMENTAL SETUP

N (τ )dτ = −∞, k0

for some k0 ∈ (k 0 , ∞), where k 0 ∈ R. It is easy to see that Definition 2.1 holds true for some k0 ∈ (k 0 , ∞) iff it is valid for all k0 ∈ (k 0 , ∞) [2]. III. S TABILITY A NALYSIS OF F RACTIONAL UAS This section discusses the asymptotic stability of two Fractional scalar systems by using the method of fractional UAS: (I) Fractional dynamics with integer order control strategy, and (II) Fractional dynamics with fractional control strategy. The mathematical results presented here are from [1].

Based on the theoretical results in section III-A the experiments have been performed on a DC motor, with both integer and fractional order UAS control strategy. The fractional order UAS was tested on a hardware-in-the-loop setup. Fig. 1 shows the details of the system and Fig. 2 shows the actual setup. The diagram shown in Fig. 1 uses the control strategy as in (8). The plant used is a DC motor, which is interfaced to the Quanser Q4 board through a combined AD/DA and power amplifier unit [14]. The saturation bound of the plant input is ±5V . The Nussbaum function used here is the Mittag-Leffler function, represented by the ’s-function’ block in Fig. 1. The Mittag-Leffler function is coded in ’C’ as a c-mex ’*.dll’ file since the existing Simulinkr custom function block is not

Fig. 3.

Fig. 1.

Integer order UAS - Squarewave tracking

Hardware in-the-loop fractional UAS control scheme

Fig. 4.

Integer order UAS - Sinewave tracking

Fig. 4 it is clear that the integer order UAS does not track a sinusoidal reference well. It has a slow response, and it fails to track the reference midway, after which the error is around 0.1275. It can be seen from Fig. 5 that for α = 0.6 Fig. 2.

Actual hardware in-the-loop setup

yet supported by real time workshop (RTW). Coding it in ’C’ makes it operate much faster than a normal MATLABr function call. The blocks to the right of the fractional derivative block are used to change the type of the derivative from Riemann-Liouville form to the Caputo [7] form. The error e(t) is used to obtain the function k(t), which is fed to the MittagLeffler function to produce the Nussbaum function N (k(t)). Note that the parameter α used for the Mittag-Leffler function is different from the fractional order α used in the control loop. All the parameters a, b, c and a2 are initially set to 1. The effect of varying these parameters are explored later in the paper. VI. E XPERIMENTAL RESULTS The experiments are first conducted with the integer order UAS control law, they are then repeated with the fractional order UAS and the results are compared. As can be seen from Fig. 3, the integer order UAS has a finite steady state error, which is quite big. Also it has a slow response time. From

Fig. 5.

Fractional order UAS - Squarewave tracking, α = 0.6

the fractional order UAS performs really well. It has a fast response and the steady state error is of the order of 0.02. The control effort is highest when the squarewave switches. This behavior is obvious since the error is the highest at that point. From the figures 5, 6, and 7 it can be seen that as α decreases, the steady state error increases. As α increases a

Fig. 6.

Fractional order UAS - Squarewave tracking, α = 0.1

Fig. 9.

Fractional order UAS - Sawtooth wave tracking, α = 0.5

is beyond most of the time. When the functions p saturation πk 2 k cos( |k|) and cos( 2 )ek are used as Nussbaum functions, they do not cause extreme control effort saturation, but they produce a lot of output noise. When the above mentioned Nussbaum functions are used to track a sinusoidal reference, a lot of noise is introduced initially and the error is a lot higher than observed in the previous cases.

Fig. 7.

Fractional order UAS - Squarewave tracking, α = 0.75

lot of noise is introduced into the system and the settling time increases.

Fig. 10.

Fractional order UAS - Nussbaum function = k2 cos(k)

B. Other experiments

Fig. 8.

Fractional order UAS - Sinewave tracking, α = 0.5

From Fig. 8 it is seen that the fractional order UAS can track a sinewave reference. The error is in the range of 0.02 to 0.04. Fig. 9 shows that the fractional order UAS can track a sawtooth reference. The steady state error is around 0.005. A. Effect of different Nussbaum functions on the fractional order UAS From Fig. 10 it is clear that using k 2 cos(k) as the Nussbaum function causes noise in the output and the control effort

Various experiments have been performed for different values of the parameters b, c and a2 . Different functions like ˙ k(t) = |y(t)|, y 4 (t) or λy γ (t), γ ∈ R and λ is a constant, have been tried. Increasing the power of y or certain values for λ affect the performance adversely, whereas some other cases are not the best but are operational. A change in sign of the R.H.S. terms of the control input in equation (8) do not have bad effects but it increases the steady state error. The same experiments have been tried out with a high gear and low gear configuration for the motors. It is observed that the gear configuration does not affect the performance of the fractional order UAS. From Fig. 11 it is observed that changing the second part ˙ of equation (8) to k(t) = |y(t)| has no drastic effect, but the steady state error and the control effort increase by a small ˙ amount. However if it is changed to k(t) = y 4 (t), the output

Fig. 11.

Fig. 12.

˙ k(t) = |y(t)|

u(t) = +N (k(t))y(t) + (cb)−1 a2 0 Dtα y(t)

Fig. 13.

Fig. 14.

Squarewave tracking -

Squarewave tracking -

a2 cb

a2 cb

= 0.5

= −0.5

becomes very noisy and a big steady state error is introduced; also the control effort is always beyond saturation, which is not good. If the sign of the control effort in equation (8) is changed, then as observed in Fig. 12 the system is still operational due to the Nussbaum functions ability of searching. However, the steady state error and the control effort increase. C. Effect of varying b, c and a2 In the previous experiments the values of the variables b, c and a2 are all taken equal to one. Thus it is obvious that the fractional order UAS works when the co-efficient acb2 of the second term in (8) equals one. The following experimental results explore the performance of the fractional order UAS when the value of acb2 is changed. From Fig. 13 it is seen that the fractional order UAS works well when acb2 = 0.5. , it is found that The performance does not deteriorate when acb2 = 2. However, when acb2 = −0.5 the performance is a bit slower, i.e. the settling time is a bit higher as seen in Fig. 14. Also in this case, the error is a little larger in the beginning, but it is seen to reduce with time. The system however does not follow the squarewave as well as the previous cases because each cycle of the squarewave has a certain small amount of disturbance in the beginning. D. Very low speed regulation and external noise Experiments were performed when the reference signals were set to very low levels, and with external noise injected

Fig. 15.

Squarewave tracking ±0.1rad/s

in the loop. The results are as follows. Figure 15 shows that, the fractional order UAS control scheme works well even for a low reference of 0.1 rad/s. The control effort never goes beyond the saturation limit and the output is not noisy, any initial error decreases with time. For a reference of 0.05 rad/s initially the error is larger than the previous case but it decreases with time. The output is a bit noisier than the previous case since frictional resistance and sensor noise play a vital role when tracking such low speeds. Tracking a reference lower than 0.05 rad/s for this particular motor is not possible. Fig. 16 shows the case when additional band-limited white

X. A PPENDIX A. Mathematical results The following theorems, corollaries and lemmas have been presented from [1], for full length proofs consult [1].

1) Results for section III-A: Corollary 10.1: There exist t1 ∈ [0, t] and Mmax = max{[a − bN (k(t))]|t ≥ t1 } < 0 such that 0 ≤ x(0)x(t) ≤ x2 (0)eMmax t .

emax t . Theorem 10.2: For system (9), 0 ≤ y(0)y(t) ≤ y 2 (0)eM Theorem 10.3: In (8), u(t) = −N (k(t))y(t)−(cb)−1 a2 0 Dtα y(t) tends to zero as t → ∞. B. Code for Mittag-Leffler ’s-function’ block

Fig. 16.

Squarewave tracking ±1rad/s, external noise added

noise is injected into the loop. Even then, the fractional UAS does a good job. The error and the output is similar to the results obtained without noise and the minimum steady state error obtained is of the order of 0.01. However, when disrupted by noise, some spikes are seen in the error which quickly settle down. The maximum steady state error is around 0.275 which decreases promptly to 0.06. VII. O BSERVATIONS FROM THE TESTS The fractional order UAS definitely works better than the integer order case. The performance deteriorates when the fractional order α is made too less or too large. It tracks a squarewave and sinusoidal references with ease. The performance depends on the selection of a proper Nussbaum function. The fractional order UAS works with different values of acb2 . Using different functions and different gains on the ˙ R.H.S. of the equation k(t) = y 2 (t), have a profound impact on the performance. Changing the sign of the terms on the R.H.S. of equation (8) generally changes the steady state error by a sizeable amount. The fractional order UAS performs well even when tracking a reference of very small magnitude or when additional noise is induced deliberately into the loop. VIII. C ONCLUSION When properly tuned, with the right parameters, the fractional order UAS works better than the integer order one. The experiments have been performed on a DC motor, hence its application is not restricted to models with fractional dynamics. The response of the fractional UAS is fast, and it works remarkably well even when tracking sinusoidal and sawtooth references. IX. F UTURE WORK Finding the optimal values for the parameters a, b, c, a2 and α is a definite direction for future work. Also the reason, as to why the output is different for different Nussbaum functions, and why the fractional order UAS works best when a MittagLeffler function is used as a Nussbaum function, need to be researched. An explanation also needs to be found for the fractional order UAS’s ability to track sinusoidal references.

The ’C’ code for the Mittag-Leffler function and it’s c-mex ’*.dll’ version can be found at [15]. The gamma function used here is obtained from [16]. The gamma function obtained from [16] is verified to be accurate. Also the ’C’ version of Mittag-Leffler function is verified against its MATLABr version. The output from both is exactly the same and accurate in the range of interest for this experiment.

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