Tuning of the half-order robust PID controller

Tuning of the half-order robust PID controller dedicated to oriented PV system Wojciech Mitkowski and Krzysztof Oprzedkiewicz AGH University of Science and Technology, Faculty of Electrotechnics, Automatics, Informatics and Biomedical Engineering, Dept of Automatics and Biomedical Engineering, Al. A. Mickiewicza 30, 30059 Krakow, Poland

Abstract. In the paper tuning rules for half - order PID controller dedicated to control an oriented PV system were presented. The plant is described with the use of interval transfer function. Results were by simulations depicted. Keywords: Fractional order PID controller, PV systems, minimal-energy control, interval systems, robust control

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An introduction

Problems of modeling and control for dynamic systems described with the use of non-integer models were presented by many Authors, for example: [23], [24], [2], [4] , [5], [6], [19], [25]. Problems of control fractional order, interval systems were presented for example in [20], [21], [22]. The another area of non-integer order calculus in control are non integer order controllers. They are often applied in many control systems. This problem was presented for example in [24] or [22]. It is caused by a fact, that this class of controllers assures better control performance, than traditional integer order control. In this paper a minimal-energy control of an uncertain-parameter oriented PV system with the use of half-order PID controller is considered. The term ”half order” describes orders of derivative and integral actions: they both are equal 0.5. For this system an analysis of BIBO (Bounded Input Bounded Output) stability is possible with respect to uncertainty of plants parameters. In the paper the following problems will be presented: – – – – – –

An oriented PV system and its model, Closed-loop control system with fractional order PID controller, Stability analysis for closed-loop system, ORA approximation for both parts of controller, PID tuning methods for the considered system, An example.

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Wojciech Mitkowski and Krzysztof Oprzedkiewicz

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An oriented PV system and its model

Let us consider a moving part of an oriented PV system shown in figure 2. The most simple scheme of this plant is a DC motor with gearbox, considered by many Authors (see [1], p. 453, 459; see [22], p. 121) shown in figure 1.

R

i (t ) L

 (t ) x1 ( t )

u (t ) I  const .

J

Fig. 1. A DC electric drive as a model of moving part of the oriented PV system

The exact description of the plant we deal with can be found in [15], [12]. The most simple model of the plant shown in figure 1 is a transfer function with interval parameters: k (1) G(s, q) = Ti s where: k > 0 and Ti denote interval parameters of the PV, assembled in vector q ∈ Q defined as follows: Q = {q = [k, Ti ] : k ≤ k ≥ k Ti ≤ Ti ≥ Ti } ⊂ I(R2 )

(2)

Vertices (corners) of the set Q are defined as underneath: qll = [k, Ti ] qlh = [k, Ti ] qhh = [k, Ti ] qhl = [k, Ti ]

(3)

Vector q ∈ Q describes parameters of the plant, changing during work of the system outdoor in extremally different atmospheric conditions (summer and winter, with and without snow, etc.). Additionally - these parameters have different values for moving up and moving down the PV. Exemplary values of these parameters are given in an example.

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Fig. 2. An oriented PV system

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Closed-loop control system with fractional order PID controller

The closed-loop control system with half-order PID controller is shown in figure 3. In scheme shown in figure 3 G(s, q) denotes the plant described with the use of transfer function (1), Gc (s) denotes a fractional-order PID controller, described as underneath: kI Gc (s, p) = kp + α + kD sβ (4) s where kp > 0, kI > 0 and kD > 0 describe the proportional, integrating and derivating actions of the controller, α and β are fractional orders of the integration and derivation actions. The controller parameters are assembled in vector p: 3 p = [kP , kI , kD ] ∈ P ⊂ R+ (5) where P (q) is a set of all vectors p possible to technical realization. It can be also described by intervals. The transfer function of the open loop system shown in figure 1 is equal: Go (s, p, q) =

kkD sα+β + kkP sα + kkI Ti sα+1

(6)

The transfer function of the whole closed-loop system defined as: Gz (s) = is expressed as follows:

Y (s) R(s)

Gz (s, p, q) =

kkD sα+β + kkP sα + kkI Ti sα+1 + kkD sα+β + kkP sα + kkI

(7)

The above transfer function will be applied to stability analysis and optimal tuning of the fractional - order PID controller.

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Wojciech Mitkowski and Krzysztof Oprzedkiewicz

R(s)

E(s)

-

Gc(s)

U(s)

G(s,q)

Y(s) )

Fig. 3. A closed-loop control system for the considered PV system

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Robust stability analysis of control system

The characteristic polynomial of the control system we deal with is a function of vectors p and q and it is equal: W (s, p, q) = Ti sα+1 + kkD sα+β + kkP sα + kkI

(8)

Now let us make an important assumption about fractional orders of integrating and derivating actions in controller: assume, that α = 0.5 and β = 0.5. This assumption will make the asymptotic stability analysis of the system much more simplier. Additionally, results of simulations show, that stability conditions obtained for those fixed values can be generalized for broad range of fractional orders α and β. It will be shown in the Example. After the above assumption the polynomial (8) reduces to the form: W (s, p, q) = Ti s1.5 + kkD s + kkP s0.5 + kkI

(9)

If we replace: λ = s0.5 , then the polynomial (8) reduces to the following integer order quasi-polynomial: Wλ (λ, p, q) = Ti λ3 + kkD λ2 + kkP λ + kkI

(10)

The BIBO stability of quasi polynomial Wλ (λ, p, q) described by (10) determines the BIBO stability of polynomial (9) with respect to the following conditon (see for example [2], p. 21 and 22): Theorem 1. The system (10) is BIBO stable (Bounded Input Bounded Output) if and only if:

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|arg(λi )| > α π2 ∀i = 1...N where λi denotes i-th root of characteristic polynomial W (sα ) Notice, that for α = 0.5 the stability area covers also the part of right complex semiplane and it covers the whole stability area for integer order system with α = 1. Next, the PD controller (kI = 0) makes the system structurally stable, but the PI controller (with kD = 0) is not able to assure the asymptotic stability, because it causes the lossing one of parameters the quasi polynonial (10). The system with PI controller will be stable, but not asymptotically stable. The quasi-polynomial Wλ (λ, p, q) is an interval polynomial and to test their stability the Charitonov theorem can be applied (see for example [3]). Next - the stability of quasi-polynomial(10) implies the stability of fractional-order polynomial (9). The Hurwitz array H(p, q) for the quasi-polynomial (10) is also a function of vectors p and q and it has the following simple form:   kkD kkI 0 (11) H(p, q) =  Ti kkP 0  0 kkD kkI From Hurwtiz criterion we at once formulate the sufficient and necessary condition of asymptotic stability the quasi-polynomial (10), expressed with the use of both vectors p and q: k (kkD kP − kI Ti ) > 0 ⇐⇒ kI