STABILIZATION USING FRACTIONAL-ORDER PI AND PID CONTROLLERS
Serdar E. HAMAMCI Inonu University, Engineering Faculty, Electrical-Electronics Eng. Dept., 44280 Malatya TURKEY. (e-mail:
[email protected]; tel: +90 422 3410010-ext.4500)
Abstract This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PI and PI D controllers. It is based on plotting the global stability region in the (kp, ki)plane for the PI controller and in the (kp, ki, kd)-space for the PI D controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.
Keywords: fractional-order control, fractional-order PI controller, fractional-order PID controller, global stability region, stabilization.
1
1. INTRODUCTION The fractional-order system is a dynamic system represented by differential equations where the orders of derivatives can take any real number, not necessarily integer number. Expanding calculus to fractional orders has been known since the development of the regular calculus, with the first reference probably being associated with correspondence between Leibniz and L’Hospital in 1695. The earliest systematic studies about fractional calculus were made by some researchers such as Liouville (1832), Holmgren (1864) and Riemann (1953), to name a few [1]. As one of fractional-order calculus’s applications, fractional-order control was introduced by Tustin for the position control of massive objects fifty years ago [2]. Some other researches were fulfilled around 1960’s by Manabe [3]. However, fractional-order control was not incorporated into Control Engineering mainly due to the lack of sufficient mathematical knowledge and the limited computational power available at that time. In last few decades, researchers found that the fractional-order differential equations could model various systems more adequately than integer-order ones and provide an excellent tool for describing dynamic processes [4, 5]. On these studies, Oustaloup [6] introduced the fractional-order algorithms for the control of dynamic systems by the CRONE (French abbreviation for CommandeRobuste d’Ordre Non Entier) method over the PID controllers which have been dominating industrial controllers. More recently, Podlubny [1, 7] has proposed a generalization of the PI and PID controllers, namely the PI and PI D controllers, involving an integrator of order and differentiator of order (the orders and may assume real noninteger values). Podlubny has also demonstrated the better response of these types of controllers, in comparison with the classical PI and PID controllers, when used for the control of fractional-order systems. A frequency domain approach by using fractional PID controllers has been also studied in [8]. Further research activities are running in order to develop new tuning rules for fractional controllers, studying previously the effects of the non integer order of the derivative and integral parts to design a more effective controller to be used in real-life models [9]. Some of these techniques are based on an extension of the classical PID control theory. To this respect, in [10] a more flexible tuning strategy and therefore an easier achieving of control requirements with respect to classical controllers has been presented with the extension of differentiation and integration order from integer to non integer numbers. In [11] an optimal PI D controller based on specified gain and phase margin with a minimum ISE criterion has been designed by using a differential evolution algorithm. Since the minimal requirement for the controllers is to make the system stable, it is desirable to know the complete set of the stabilizing PI or PID controller parameters for a given plant before controller design and tuning. Many important results have been recently reported on computation of all stabilizing PI and PID 2
controllers after the publication of the work by Ho et al [12] which is based on generalized version of the Hermite-Biehler theorem. However, this method needs sweeping over the proportional gain to find all stabilizing PI and PID controllers which is a disadvantage of the method. An alternative fast approach to this problem based on the use of the Nyquist plot has been given in [13]. A parameter space approach using the singular frequency concept has been given in [14] for design of robust PID controllers. More direct graphical approaches based on frequency response plot have been given in [15, 16]. On the other hand, controller design for the systems containing uncertainties has been studied in [17, 18]. More recently, a controller design technique based on the stability boundary locus has been given in [19, 20]. However, the stabilization problems considered in these methods completely deal with systems dynamics whose behaviors are described by integer-order differential equations. The formulation, numerical scheme and numerical results for the stabilization of the fractional-order systems presented in this paper are attempts to fill this gap. In this paper, an approach to obtain of stabilizing fractional-order PI controllers for fractional-order systems in the parameter plane, i.e. (kp, ki)-plane, is given. The problem of the determining all stabilizing values of the parameters of a PI controller is examined that enables the controller parameters to stay within a general stability region. Using this approach, a very fast and simple way of calculating the stabilizing values of PI controllers is provided. An expansion of the method to obtain all stabilizing values of the parameters of a PI D controller, namely kp, ki and kd, in the (kp, ki, kd)-space is also given. The presented method is important to observe the changing of general stability region for any given system when the values of the parameter orders and are changed. Thus, it is seen from the simulation studies that the fractional-order controllers can provide the bigger stability regions than the integer-order ones. The proposed method is also used to stabilize the systems which have a set of disconnected stability regions. Furthermore, this approach provides several considerable advantages such as it can be applied to the fractional-order systems including time delay and also fractional-order chaotic systems. To the best knowledge of author, these problems has not yet analyzed for the PI and PI D controllers.
2. FRACTIONAL ORDER SYSTEMS AND PI / PI D CONTROLLERS 2.1. Fundamentals of Fractional Calculus and Fractional Order Systems Fractional calculus deals with derivatives and integrals to an arbitrary order (i.e., rational, irrational or even complex). Since its foundation, this area of mathematics has been the subject of several approaches leading to several definitions of fractional derivatives and integrals [1]. 3
Definition 2.1. A fundamental operator aDtγ , a generalization of differential and integral operators, is introduced as follows: d γ / dt γ aDtγ = 1 t ( dτ ) −γ a
ℜ(γ ) > 0 ℜ(γ ) = 0
∫
(1)
ℜ(γ ) < 0
where is the fractional order which can be a complex, and a is the constant related to initial conditions. In (1), ℜ(γ ) denotes the real part of the fractional order. Oldham and Spanier [21] called Equation (1) differintegral operator since it unifies on a single operator the notions of integral and derivative. The most fundamental definition of a fractional derivative and integral of order is given by Grünwald–Letnikov definition: aDtγ
1 f (t ) = lim γ h →0 h
[(t − a ) / h ]
∑ j =0
γ ( −1) j f (t − jh) , j
(2)
γ Γ (γ + 1) = j Γ ( j + 1) Γ (γ − j + 1)
(3)
where +(x) is the well known Euler’s gamma function, h is the time increment, f(t) is the applied function and [.] is a flooring-operator. This definition reveals that while integer-order derivatives imply a finite series, the fractional-order derivatives require an infinite number of terms. In the analysis and synthesis of automatic control systems, the Laplace transform method is used commonly. The Laplace transform of the differintegral operator aDtγ is given by the expected form:
£{ D 0
γ t
∞
f (t )}= ∫ e
− st
γ 0 Dt
f (t )dt = s γ F ( s ) −
n −1
∑s m=0
0
m
( −1) j 0 Dtγ − m−1 f (t ) |t =0
(4)
where F ( s ) = £ {f (t )} is the normal Laplace transformation, n is integer that satisfies n-1 < ≤ n. If γ − m −1 0 Dt
f (t ) t =0 = 0,
m = 0, 1, 2,......, n − 1
(5)
then
£ {0 Dtγ f (t )}= s γ F ( s )
(6)
Furthermore, its Fourier transform can be exactly obtained by substituting s with j& in its Laplace transform just like its integer order counterpart. Definition 2.2. A fractional-order system is defined by the system which is better described by the fractional-order mathematical models. Contrary to the conventional approach, the fractional-order system 4
has the transfer function of an arbitrary real order. Consider the fractional-order transfer function given as the following expression: n
β β β β N ( s) bn s n + bn −1 s n −1 + ......... + b1 s 1 + b0 s 0 = = G ( s) = D ( s) a n s α n + a n −1 s α n −1 + ......... + a1 s α1 + a 0 s α 0
∑b s
βi
∑a s
α1
i
i =0 n
(7)
i
i =0
where ai, bi, β n > ..... > β 1 > β 0 ≥ 0 and α n > ..... > α 1 > α 0 ≥ 0 are arbitrary real numbers. In the time domain, G(s) corresponds to the (n+1)-terms inhomogeneous fractional-order differential equation n
∑
ai D α i y (t ) =
i =0
n
∑b D i
βi
u (t )
(8)
i =0
where y(t) is the output and u(t) is the input of the plant of Equation (7).
2.2. Fractional Order PI and PI D Control Systems A general SISO fractional-order control system is shown in Figure 1. In this figure, y is the output, r is the reference input, e is the error and u is the control signal. G(s) is the transfer function of the system and C(s) is the transfer function of the controller in the type of fractional-order PI or PI D controller. The system to be controlled with these types of the controllers can be both fractional-order and integer order. The transfer function of the controller has the form C (s) =
k U ( s) = k p + λi + k d s µ E ( s) s
(9)
where and are the fractional orders whose values are in the range of (0,2). If λ ≥ 2 or µ ≥ 2 , the controller is transformed to higher order structure which is the different form than the PID structure. If the differential element is not considered, the PI D controller is turned into the PI controller given as C (s) =
k U (s) = k p + λi E ( s) s
(10)
Taking λ = 1 and µ = 1 in Equation (9), it is obtained a classical PID controller. λ = 1 and µ = 0 give a PI controller, λ = 0 and µ = 1 give a PD controller, and λ = 0 and µ = 0 give a gain. All these classical types of PID-controllers are the particular cases of the fractional PI D controller [7]. Thus, the PI D controllers
r
e _
u
C(s) controller
G(s)
y
plant
Figure 1. A general SISO fractional-order control system structure. 5
can be considered as the generalization of the conventional PID controllers because of involving an integrator of order and a differentiator of order . However, the PI D controller is more flexible and gives an opportunity to better adjust the dynamical properties of a fractional-order control system.
3. STABILIZATION USING FRACTIONAL-ORDER PI CONTROLLER Consider the unity feedback fractional-order control system shown in Figure 1, where G(s) is the plant to be controlled given in Equation (7), and C(s) is a PI controller which has form of Equation (10). The problem is to compute the parameters of the fractional-order PI controller stabilizing the plant of Figure 1. The output of the control system is given by y=
C ( s)G ( s) r . 1 + C ( s)G ( s)
(11)
Definition 3.1. The denominator of Equation (11) is described as fractional-order characteristic polynomial (FOCP) of the closed loop system. From Equations (7), (10) and (11), the FOCP of the control system is obtained as P( s; k p , k i , λ ) = s λ D( s ) + (k p s λ + k i ) N ( s) =
∑ [a s n
i
(α i + λ )
]
+ k p bi s ( β i + λ ) + k i bi s β i .
i =0
(12)
For a given fractional-order PI controller parameters , kp and ki, the closed-loop system is said to be bounded-input bounded-output (BIBO) stable if the quasipolynomial P( s; k p , k i , λ ) has no roots in the closed right-half of the s-plane (RHP). The stability domain S in the parameter space
P with , kp and ki
being coordinates is the region that for ( , kp, ki)∈S the roots of quasipolynomial P( s; k p , k i , λ ) all lie in open left-half of the s-plane (LHP). Therefore, the determination of stability domain S is an important task for the design of the PI controllers. In general, the characteristic polynomial can be represented as k
P( s ) = pk s qk + p k −1s qk −1 + ......... + p1 s q1 + p0 s q0 = ∑ pi s qi
(13)
i =0
where pi are the coefficients and qi are the fractional orders of the FOCP. Here, all or some of the coefficients pi are the function of kp and ki depending on the order of N(s) and D(s) polynomials. The boundaries between the stability and instability domains in the parameter space
P
are defined by the
following three parts [20]: (i) Real Root Boundary (RRB): A real root crosses over the imaginary axis at s=0. Thus, the real root boundary is obtained from substituting s=0 in P(s) of Equation (13). This boundary can be 6
determined as p0=0 if only the value of the smallest order of the FOCP (q0) is zero, that is s q0 = 1 . (ii) Complex Root Boundary (CRB): A pair of complex roots crosses over the imaginary axis at s=jω. Therefore, Equation (13) becomes unstable which means that the real and the imaginary parts of this equation become zero simultaneously. (iii) Infinite Root Boundary (IRB): A real root crosses over the imaginary axis at s=∞. Thus, the infinite root boundary can be characterized by taking pk=0 from Equation (13). In applying the above descriptions to the FOCP in Equation (12), it follows from Part (i) that the RRB is determined as ki = 0
(14)
for s β 0 = 1 in the transfer function of the plant in Equation (7). This boundary is obtained by substituting s β 0 = 1 and s = 0 into Equation (12) and equating it to zero. To construct the CRB, we substitute s=j& into Equation (12) to obtain
∑ [a ( jω ) n
P(ω ; k p , k i , λ ) =
i
(αi + λ )
]
+ k p bi ( jω ) ( βi +λ ) + k i bi ( jω ) βi = ℜ{P( s)}+ jℑ{P( s )} = 0
i =0
(15)
where ℜ{P (s)} and ℑ{P(s)} denote the real and the imaginary part of the FOCP, respectively. The noninteger power of a complex number (σ + jω )γ can be calculated by
ω ω (σ + jω ) γ = (σ 2 + ω 2 ) 0.5γ cos(γ tan −1 ) + j sin(γ tan −1 ) σ σ
(16)
where 1 is the real part, ω is the imaginary part and is the fractional order of the complex number. Using Equation(16), the term of j γ which is required for Equation (15) can be expressed by j γ = cos(γ
π π ) + j sin(γ ) . 2 2
(17)
Hence, P(jω) can be written as P(ω ; k p , ki , λ ) =
n
∑ a ω i
π π cos[(α i + λ ) ] + j sin[(α i + λ ) ] 2 2
(α i + λ )
i =0
+ k p biω
π π π π β cos[( β i + λ ) ] + j sin[( β i + λ ) ] + k i biω i cos( β i ) + j sin( β i ) 2 2 2 2
( βi +λ )
(18)
Then, equating the real and imaginary parts of P(M&) to zero, one obtains n
i
i
π
π
π
∑ aiω (α +λ ) cos[(αi + λ ) 2 ] + k pbiω ( β +λ ) cos[( βi + λ ) 2 ] + kibiω β cos( βi 2 ) = 0 i
i
i =0 n
π
π
∑ aiω (α +λ ) sin[(αi + λ ) 2 ] + k pbiω ( β +λ ) sin[(βi + λ ) 2 ] + kibiω β i
i =0
7
i
sin( β i
π ) =0 2
(19)
Finally, by solving the 2-dimensional system of Equation (19) the PI controller parameters are obtained as kp =
ki =
A1 (ω ) B2 (ω ) − A2 (ω ) B1 (ω ) π B12 (ω ) + B22 (ω ) ω λ sin( λ ) 2 1
(20)
A2 (ω ) B3 (ω ) − A1 (ω ) B4 (ω ) π B12 (ω ) + B22 (ω ) ω λ sin( λ ) 2 1
(21)
where n n n π π π A1 (ω ) = ∑ aiω (α i +λ ) cos[(α i + λ ) ] , A2 (ω ) = ∑ aiω (α i +λ ) sin[(α i + λ ) ] , B1 (ω ) = ∑ biω βi cos( β i ) 2 2 2 i=0 i =0 i=0 n
B2 (ω ) = ∑ biω βi sin( β i i =0
n n π π π ) , B3 (ω ) = ∑ biω ( βi +λ ) cos[( β i + λ ) ] , B4 (ω ) = ∑ biω ( βi +λ ) sin[( β i + λ ) ] (22) 2 2 2 i =0 i =0
Changing ω from 0 to ∞, a CRB curve is constructed in the (kp, ki)-plane using Equations (20) and (21). The IRB can be only determined for α n = β n in the transfer function of the plant in Equation (7). Then the stability region obtained by Equations (20) and (21) is bounded with the line of k p = − an bn .
(23)
This is found by equating the term of the largest order of FOCP in Equation (12) to zero. Mostly, since the order of denominator is larger than the order of the numerator of the plant transfer function, the IRB does not exist. The CRB curve divides the entire parameter space (kp , ki) into stable and unstable regions when the RRB and IRB lines corresponding to the special conditions do not exist. The stable region, which is called global stability region, can be determined by choosing a test point within each region. The characteristic polynomial belonging to the stable region have no RHP roots until the characteristic polynomial of the unstable region have a certain number of RHP roots. This procedure is repeated for the values of in the range of (0, 2) until the biggest stability region is obtained. In some cases, it can be found more than one stable regions and/or unstable regions. These cases are given as follows: 1. The case of s β 0 = 1 : The CRB curve crosses the RRB line at one or more points. At these points, the frequency values of CRB curve are determined by substituting ki = 0 in Equation (21). 2. The case of αn=βn: The CRB curve and the IRB line intersect at one or more points. The intersection points and its frequency values for the CRB can be found by equating Equation (20) with (23). 3. The case of s β 0 = 1 and αn=βn: This case is a combination of two cases above. The CRB curve crosses 8
both the RRB line and the IRB line at many points. In addition to this, the RRB and IRB lines intersect at the point of (kp, ki)=(-an/bn, 0). As a result of these intersections, too many regions can be obtained. The global stability region is determined by testing each region. Sometimes it can be found two or more disconnected stable regions. An important advantage of the proposed method is that an extra calculation is not needed for determining the disconnected stable regions. The presented stabilization algorithm for the PI controller is summarized as follows: Step 1. Use Equations (20) and (21) to obtain the equations of kp and ki in terms of λ for the CRB curve. Step 2. Investigate the presences of the RRB and IRB lines. Step 3. For any λ value; a) Obtain all regions plotting the IRB line, RRB line and the CRB curve in the same (kp, ki)-plane, b) Determine the global stability region by checking each regions using the arbitrary test points. Step 4. Find a set of global stability regions using different values of λ∈(0, 2) for the PI controller. Step 5. Choose a value of λ providing the biggest global stability region.
Example 3.1: Consider the fractional-order transfer function of the heat solid (electrical radiator) [22] G ( s) =
1 as + 0.598
(24)
α
where a=39.69 and α=1.26. The objective of the design is to obtain the global stability regions including (kp, ki) points which make the closed loop characteristic polynomial stable. From Equations (20) and (21), the equations for kp and ki parameters corresponding to CRB curve are given as follows
π π π k p = − 39.69ω 1.26 sin[(1.26 + λ ) ] + 0.598 sin( λ ) / sin( λ ) 2 2 2 ki = (36.4257ω 1.26 +λ ) / sin( λ
π ) 2
(25)
(26)
The global stability region obtained by Equations (25) and (26) is bounded with the RRB line because of s β 0 = 1 . For the simplest case =1, the global stability region is obtained for the integer-order classical PI controller. Taking =1 in Equations (25) and (26), the kp and ki values are determined by k p = 15.7628ω 1.26 − 0.598 ki = 36.4257ω 2.26 9
60
ω=1.25
50
ω=1
40
CRB curve 30
ω=0.73
ki 20 10
ω=0
Global
ω=0.44
stability region
0 -10
0
5
10
RRB line 15 kp
20
Figure 2. The global stability region for the integer order PI controller ( = 1).
60 50 40
λ=0.8
30
λ=0.9 λ=1
ki 20
λ=1.2 10
λ=1.4
0 -10
RRB line 0
5
10
kp
15
20
Figure 3. The stability boundary curves for the PI controller ( =0.8, 0.9, 1, 1.2, 1.4).
Using these equations, the CRB curve and the RRB line are plotted in the (kp, ki)-plane as shown in Figure 2. It can be observed from this figure that the entire parameter plane is divided into three regions (the top of CRB, between CRB and RRB, under the RRB). By checking these regions using the arbitrary test points, the global stability region enclosed by the CRB curve and the RRB line which is the shaded region shown in Figure 2 is determined. All (kp, ki) points in the shaded region constitute the stable characteristic polynomials for the plant of Equation (24). Hence, the designer can make a decision about the choosing of the controller parameters simultaneously. Figure 3 shows the global stability regions obtained using some different values for the fractional-order PIλ controller. As can be seen from this figure, the small values 10
2.5
λ=0.8
1.5
ki=0.5 ki=5 ki=194.11
2
y(t)
y(t)
ki=2 ki=50 ( kp=10 )
1.5
1
λ=1.4 λ=1 λ=0.8 λ=0.4
0.5
1 0.5
0
0 0
10
20
30
40
t
50
0
(a)
10
20
30
40
t
50
(b)
Figure 4. a) The step responses for different values of for kp=15 and ki=2, b) The step responses for different values of (kp, ki) for PI0.8 controller.
of give the larger global stability regions than the big values of . For a fixed point of (kp, ki), for instance kp=15 and ki=2, the unit step responses for different values of are shown in Figure 4a. It is seen from this figure that the fast response and small overshoot can be obtained for small values of . Step responses of the PIλ control system when kp is chosen as 10 and ki is changed from 0.5, 2, 5, 50 to 194.11 can be shown in Figure 4b for =0.8 which gives bigger global stability region than the other values. From this figure, it is seen that the value of ki is increased from 0 to boundary value, ki=194.11, the control system has more oscillatory response. If the ki is bigger than the boundary value or smaller than zero, the control system is unstable. As evidenced by the results given in Figures 3 and 4, it can be concluded that the proposed approach is simple and reliable method for stabilizing of the fractional-order systems using PIλ controllers.
Example 3.2: Consider the control system of Figure 1 with a fractional-order plant transfer function G ( s) =
s 3.8 + 2 s 2.8 + 39 s1.9 + 48s1.1 − 4 s 5 + 2s 4.1 + 31s 3.1 + 35s 2.2 + 49s 0.9 + 92
(27)
to be controlled by a PI controller. In applying the proposed approach to find the global stability regions, the CRB curve and the RRB line which are divided the entire parameter space to eight regions, namely R1, R2, …, R8 is given in Figure 5 for the simplest case, .=1. In order to find the true stability region it is needed to choose a test point in each region. Hence, it is found that two regions denoted by R4 and R6 are the stability regions which do not include RHP poles. The other regions have at least one RHP poles, therefore, they are not stable regions. Note that the stable regions are disconnected. Figure 6 shows more clearly the global stability region which has two disconnected stability regions. 11
12
ω=4.8
ω=6.81
10 8
CRB line
6
ki
R2
4
R1
R3
2
ω=6.22
ω=1.42
0
R5
R4
RRB line
-2 -4 -5
ω=6.57
ω=0
R6 R8
ω=0.101 and 6.5 ω=0.12 and 6.35 R 7
0
5
10
kp
15
20
25
Figure 5. The boundaries in the parameter space for the integer order PI controller ( =1).
0 -0.2 -0.4 -0.6
ki -0.8 -1
Global stability region
-1.2 -1.4 -1.6 0
5
10
kp
15
20
25
Figure 6. The global stability region which has two disconnected stabilizing sets for the PI controller ( =1).
To demonstrate the effect of on the stability, the global stability regions for the various values of are plotted as shown in Figure 7. It can be seen from this figure that the smaller values of give the bigger stability regions. Furthermore, the small values connect the distinct regions. Hence, the global stability regions in a single piece are obtained. Finally, as shown by the simulations and the results given in Example 3.1 and 3.2, it can be concluded that the small values of can give bigger global stability regions than the bigger one for the fractional-order PI controllers. 12
0
0
-5
-1
-10
ki
ki -2
-15
-3
-20 -25
-4 0
20 kp
40
0
(a)
10
20
kp
30
(b)
0
0
-0.25
-0.02
ki -0.5
ki -0.04
-0.75 -1
-0.06 0
10
kp
20
0
(c)
10
20
kp
(d)
Figure 7. Global stability regions for the PI controller: a) =0.2, b) =0.6, c) =1.4, d) =1.8.
4. STABILIZATION USING FRACTIONAL-ORDER PI D CONTROLLER The method presented in Section 3 is further developed in this section to stabilize a given plant by using fractional-order PI D controller in the form of Equation (9). The closed-loop characteristic polynomial for the PI D stabilization of the fractional-order system is given by P( s) =
∑ [a s n
i
(α i + λ )
]
+ k d bi s ( βi +λ + µ ) + k p bi s ( βi +λ ) + k i bi s βi .
i =0
(28)
The lines of RRB and IRB can be obtained from the characteristic polynomial in Equation (28) k i = 0 for s β 0 = 1 RRB line: , none for s β0 ≠ 1 k d = 0 IRB line: k d = − an bn none
(29)
for (α n = β n ) or (α n > β n and µ > α n − β n ) for (α n > β n and µ = α n − β n )
.
(30)
for (α n > β n and µ < α n − β n )
It follows from Equation (30) that if αn is greater than βn, the PI D control system has three different global stability regions depending on the values of µ. This is an advantage because the designer has more freedom for choosing his desired controller. By applying the procedure given in Section 3 to construct the CRB curve, the 2-dimensional equation system with three unknowns is obtained as 13
k p B3 (ω ) + k i B1 (ω ) + k d B5 (ω ) + A1 (ω ) = 0 k p B4 (ω ) + k i B2 (ω ) + k d B6 (ω ) + A2 (ω ) = 0
(31)
where n n π π B5 (ω ) = ∑ biω ( βi +λ + µ ) cos[( β i + λ + µ ) ] and B4 (ω ) = ∑ biω ( βi +λ + µ ) sin[( β i + λ + µ ) ] . 2 2 i=0 i =0
(32)
Since the set of equations has more unknowns than equations, one of the parameters of the solution can be arbitrarily assigned. Thus, the CRB curve can be obtained with three forms: in the (kp, ki)-plane for a fixed value of kd, in the (kp, kd)-plane for a fixed value of ki and in the (ki, kd)-plane for a fixed value of kp.
For the integer-order systems, it was shown in [19, 23] that the gains ki and kd corresponding to the stabilizing region are linearly dependent each other in the (ki, kd)-plane for a fixed value of kp. In this way, the three-dimensional global stability region which is a set of the two-dimensional convex stability regions is determined. A similar approach can be taken to attain the global stability region for the fractional-order systems. If kp is arbitrarily assigned, the parameters of ki and kd can be obtained from the solution of Equation (31) ki =
1
ω
kd =
λ +µ
π sin[( λ + µ ) ] 2 1
π ω λ + µ sin[( λ + µ ) ] 2
A2 (ω ) B5 (ω ) − A1 (ω ) B6 (ω ) + k p [B4 (ω ) B5 (ω ) − B3 (ω ) B6 (ω )] B12 (ω ) + B22 (ω ) A1 (ω ) B2 (ω ) − A2 (ω ) B1 (ω ) + k p [B2 (ω ) B3 (ω ) − B1 (ω ) B4 (ω )] B12 (ω ) + B22 (ω )
(33)
(34)
It can be seen from the above equations that there are two cases depending on the values of (λ+µ) in their denominators. If (λ+µ) is equal to 2, the values of the denominators of Equations (33) and (34) are zero. Therefore, the CRB curves by using these equations are not determined. Using a graphical approach, it is possible to obtain the convex stability regions as in the stabilization of the integer-order systems. In this graphical approach, it is made use of the CRB curves obtained in the (kp, ki)-plane and (kp, kd)-plane. The controller parameters in the (kp, ki)-plane for a fixed value of kd are determined as kp =
ki =
A1 (ω ) B2 (ω ) − A2 (ω ) B1 (ω ) + k d [B2 (ω ) B5 (ω ) − B1 (ω ) B6 (ω )] π B12 (ω ) + B22 (ω ) ω λ sin( λ ) 2
(35)
A2 (ω ) B3 (ω ) − A1 (ω ) B4 (ω ) + k d [B3 (ω ) B6 (ω ) − B4 (ω ) B5 (ω ) ] B12 (ω ) + B22 (ω )
(36)
1
1
π ω λ sin( λ ) 2
and in the (kp, kd)-plane for a fixed value of ki are obtained as 14
kp =
A2 (ω ) B5 (ω ) − A1 (ω ) B6 (ω ) + k i [B2 (ω ) B5 (ω ) − B1 (ω ) B6 (ω ) ] π B12 (ω ) + B22 (ω ) ω 2 λ + µ sin( µ ) 2
(37)
kd =
A1 (ω ) B4 (ω ) − A2 (ω ) B3 (ω ) + ki [B1 (ω ) B4 (ω ) − B2 (ω ) B3 (ω )] π B12 (ω ) + B22 (ω ) ω 2 λ + µ sin( µ ) 2
(38)
1
1
Hence, two CRB curves are obtained in each plane separately. To find the global stability region using these curves, the allowable stability ranges of kp, ki and kd parameters must be determined firstly. These ranges can be determined by the intersection points of the RRB line, the IRB line and the CRB curve in the (kp, ki)-plane and (kp, kd)-plane. Then, in these ranges, two CRB curves in the (kp, ki)-plane for any two kd values and another two CRB curves in the (kp, kd)-plane for any two ki values are plotted. According to a fixed kp value in these planes, four (ki, kd) points of which two points are in one plane and two points are in the other plane are found. Thus, two lines in the (ki, kd)-plane for a fixed value of kp are computed. Finally, in conjunction with the RRB and IRB lines, the stability domain is obtained in the (ki, kd)-plane. By changing kp, the global stability region can be visualized in the three-dimensional space of (kp, ki, kd). On the other hand, if (λ+µ) is different from 2, the values of the denominators of Equations (33) and (34) are not zero. In this case, the CRB curves are obtained by using these equations directly. The solutions thus obtained can be in the form of nonconvex stability regions. When the graphical approach mentioned above is applied to the nonconvex cases, the convex results obtained can not give the true stability regions. Therefore, the values of λ and µ providing the convex stability regions are preferred. However, for only obtaining the stability ranges of the controller parameters, the graphical approach can be used for nonconvex cases. Here, it is important how the values of λ and µ are specified because these parameters have an effect on the bigness of the stability region. To attain the biggest global stability region, the stability regions which are computed by changed λ and µ in the interval of (0, 2) in the (kp, ki) or (kp, kd)-planes are obtained. In addition to this, it must be also considered the IRB in Equation (30) due to including the parameter of µ. For some systems, the global stability region can be composed of the disconnected regions. The proposed method for the PI D controller achieves to find each disconnected regions easily as in the previous section. The stabilization procedure for the PI D controller is as follows: Step 1. Investigate the presences of RRB and IRB from Equations (29) and (30). Step 2. Use Equations (33) and (34) to obtain the CRB equations for the parameters of ki and kd in terms of kp, λ and µ.
15
Step 3. Choose λ and µ giving the biggest stability region by plotting the CRB curves in (kp, ki)-plane for any kd value (or in (kp, kd)-plane for any ki value) for some values of λ∈(0, 2) and µ∈(0, 2) separately. Step 4. Find the stability ranges of the kp, ki and kd parameters using the CRB curves obtained from Equations (35), (36) and Equations (37), (38) for the values of λ and µ selected in Step 3. Step 5. Compute the stability region in the (ki, kd)-plane for any kp value in the stability range. For this, if (λ+µ)=2, use the graphical approach, else use the Equations (33) and (34) directly. Step 6. Obtain the 3-D global stability region by changing kp in the stability range. Example 4.1: Consider the temperature control of aluminum rod thermal system [24]. The transfer function of the fractional-order system has the form G ( s) =
b1 s 0.5 + b0 N ( s) = D ( s) a2 s1.5 + a1 s + a0 s 0.5
(39)
where b1 = −0.0781 , b0 = 0.1271 , a2 = 3.0641 , a1 = 0.0305 and a0 = 1 . The purpose is to obtain all the stabilizing values of the parameters of a PI D controller such that the resulting fractional-order control system is stable. The characteristic equation of the control system is derived as P( s ) = a2 s λ +1.5 + a1 s λ +1 + (a0 + k p b1 ) s λ +0.5 + k d b1 s λ +µ +0.5 + k d b0 s λ + µ + k p b0 s λ + k i b1 s 0.5 + k i b0 .
(40)
The RRB and IRB lines can be obtained from Equations (29) and (30) RRB line: ki = 0 since s β0 = 1 ,
(41)
for µ > 1 k d = 0 IRB line: k d = − a2 b1 = 39.233 for µ = 1 . none for µ < 1
(42)
In order to get the CRB curve in the (ki, kd)-plane, it is made use of Equations (33) and (34) that
π π π P(ω ) sin( µ ) − R(ω ) cos( µ ) + k p S (ω ) sin( µ ) 2 2 2 ki = − π ω S ( ) sin[( λ + µ ) ] 2 ωλ
kd = −
1
P (ω ) sin( λ
π ω sin[( λ + µ ) ] 2 µ
π π π ) + R(ω ) cos( λ ) + k p S (ω ) sin( λ ) 2 2 2 S (ω )
(43)
(44)
where P(ω ) = 0.2771ω 1.5 + 0.0781ω − 0.0899ω 0.5 , R(ω ) = 0.2393ω 2 − 0.2737ω 1.5 − 0.0039ω − 0.0899ω 0.5 and S (ω ) = −0.0061ω + 0.014ω 0.5 − 0.0162 . According to Equation (42), the PI D controller produces three different stability regions depending on the values of µ. In all cases, the RRB is the line of ki=0 as obtained in Equation (41). 16
- Case I 1, λ∈(0, 2) Here, the IRB line is kd=39.233 from Equation (42). For the simplest case, i.e. =1, the global stability region is obtained in the form of the convex sets since (λ+µ)=2. Therefore, all stabilizing PID controllers can be only computed graphically with the proposed approach plotting the stability regions in the (kp, ki)plane and (kp, kd)-plane. The stability regions in the (kp, ki)-plane for kd=39.233 and (kp, kd)-plane for ki=0 are shown in Figure 8a and b, respectively. Note that the computed CRB curves correspond to the boundary values for ki=0 and kd=39.233. Therefore, the changing ranges of the controller parameters in the stability region can be obtained easily. From this figure, the stabilizing values of controller parameters are found as kp∈[-1, 234], ki∈[0, 186.6] and kd ∈[-25.2, 39.233] for λ=µ=1. To obtain the convex set for the values of kp, the boundary curves in the (kp, ki)-plane for kd=39.233 and any kd value chosen from the interval determined above, for example kd=25, and in the (kp, kd)-plane for ki=0 and any ki value in the interval obtained, for example ki=100, are shown in Figure 9. From Figure 9a, one can obtain the equation of a straight line that
250
ω=3
CRB line
IRB line
40
200
ki
kd
ω=1.5
150 100 50
for ki=0
Stability region
ω=0
ω=5.93 -20
0 -50 -50
ω=3
0
for kd=39.233
RRB line 0
50
100 kp 150
200
250
ω=5.93
Stability region
20
ω=5
CRB line
-40 -50
300
ω=0 0
50
100 kp 150
(a)
200
250
300
(b)
Figure 8. The stability regions: a) in (kp, ki)-plane for kd=39.233, b) in (kp, kd)-plane for ki=0 ( = = 1).
250 200
ki
40
kd=39.233
(150.62, 39.233)
IRB line
kd
150
ki=100
20 (89.54, 25)
(100, 27.436)
100
kd=25
(0, 4.133)
50 -20
0 -50 -50
ki=0
0
RRB line 0
50
100 k 150 p
200
250
-40 -50
300
(a)
0
50
100 k 150 p
200
250
300
(b)
Figure 9. Determination of four points to obtain the convex stability region in the (ki, kd)-plane: a) two points in the (kp, ki)-plane, b) other two points in the (kp , kd)-plane. 17
50
IRB 40
RRB 30
Stability region
kd
for kp=100
20
10
CRB
CRB
0 -50
0
50
ki
100
150
200
Figure 10. The stability region for kp=100 ( = 1, = 1).
250 200 150
kp 100 50
0 100
0 -20
-10
ki 0
10
kd
20
30
40
200
Figure 11. The global stability region for the integer order PID controller ( = 1, = 1).
contributes to the boundary of the stability region for each value of kp. For example, for kp=100, the straight line passes through the points (ki, kd)=(150.62, 39.233) and (ki, kd)=(89.54, 25). Thus the equation of this straight line is: k d = 0.233k i + 4.1352 . Similarly, from Figure 9b, the straight line goes through the points (ki, kd)=(100, 27.436) and (ki, kd)=(0, 4.133). The equation of second straight line is k d = 0.2328k i + 4.133 . These two lines in conjunction with the RRB and IRB lines are shown in Figure 10 where the shaded region is the stability region in the (ki, kd)-plane for kp=100. By changing the proportional gain (kp) and repeating the procedure, the global stability region can be visualized in the three-dimensional plot for λ=1 as shown in Figure 11. 18
400
λ=1.8
350
200
300
100
ki 250
λ=0.4
150 100
kp 0
λ=1.4
200
-100
λ=1
-200
λ=0.1
-300
50 0 -50 -300
-400 -20 -200
-100
0
kp 100
200
0
kd
300
(a)
20
40
750
500
250
0
ki
(b)
Figure 12. a) The stability regions of some values ( =1 and kd=25). b) The global stability region for the PI0.1D controller.
It is noted that different choices of leads to different global stability regions. To achieve the suitable choice, the CRB curves for the different values of over the interval (0, 2) using a fixed value of kd, for example kd=25, are plotted as shown in Figure 12a. It is clearly seen from this figure that the choice of =0.1 is the most suitable selection because of providing the biggest CRB curve. The exact range of
controller parameters for the PI0.1D controller are found as kp∈[-374.35, 233.9], ki∈[0, 500.14] and kd∈[25.2, 39.233] by using the graphical approach. These values are greater than the values of the integer-order PID controller. Since (λ+µ)≠2 the global stability region can be obtained from Equations (33) and (34) directly. The global stability region in (kp, ki, kd)-space for the values of kp∈[-374.35, 233.9] are shown in Figure 12b. It is seen from this figure that the stability regions in (ki, kd)-plane are in the nonconvex form. - Case II >1, λ∈(0, 2) In this case, the IRB is the line of kd=0. To determine the biggest global stability region, the values of λ and
µ must be determined. For this, the CRB curves in the (ki, kd)-plane are used directly because (λ+µ)=2 for some λ and µ values. Therefore, the CRB curves which are obtained in the (kp, ki)-plane or in the (kp, kd)plane can be used. The CRB curves using Equations (35) and (36) for some values of !1 and λ∈(0, 2) are shown in Figure 13a and b, respectively. These curves are plotted for kd=-5. As can be seen from these figures that the PI0.1D1.9 controller provides the biggest global stability region. The stabilizing values of the controller parameters for this controller are found as kp∈[-72.7, 86.8], ki∈[0, 99.8] and kd∈[-70.1, 0]. According to the range of kp obtained, the global stability region is shown in Figure 14. It is seen from this figure that the global stability region is in the convex form.
19
120 100
20
µ=1.9
λ=0.1 µ=1.4
15 80
ki
λ=1.8
60
λ=0.4
40 20 0 -100
µ=1.7
ki 10
λ=1.4
µ=1.1
5
λ=1 -50
0
kp
50
0 -20
100
0
20
(a)
40
kp
60
80
(b)
Figure 13. The stability regions: a) for some λ values b) for some µ values.
80
40
kp
0
-40 0 50 -80 -70
ki -60
-50
-40
-30
-20
kd
-10
0
10
100
Figure 14. The global stability region for the PI0.1D1.9 controller.
- Case III
