ROBUST CONTROL FOR UNCERTAIN DELAY ... - Semantic Scholar
Journal of Circuits, Systems, and Computers Vol. 20, No. 3 (2011) 479499 # .c World Scienti¯c Publishing Company DOI: 10.1142/S0218126611007402
ROBUST CONTROL FOR UNCERTAIN DELAY SYSTEM
¤
RIHEM FARKHy,{, KAOUTHER LAABIDIIz and MEKKI KSOURIx Research unit on Systems Analysis and Control, National School Engineering of Tunis, BP, 37, Le Belvedere, 1002 Tunis, Tunisia y [email protected] [email protected] [email protected] Received 6 February 2010 Accepted 30 November 2010 In this paper, the Generalized Kharitonov Theorem for quasi-polynomials is exploited for the purpose of synthesizing a robust controller. The aim here is to develop a controller to simultaneously stabilize a given interval plant family with unknowing and bounded time delay. Using a constructive procedure based on HermiteBiehler theorem, we obtain all PI gains that stabilize an uncertain ¯rst-order delay system. An application example is presented for the temperature control of a heated air stream, process trainer P T 326. Keywords: Uncertain time delay system; interval plants; Generalized Kharitonov Theorem (GKT); PI controller; HermiteBiehler theorem; stability region.
1. Introduction Dead times are often encountered in various engineering systems and industry processes such as electrical and communication network, turbojet engine, chemical process, hydraulic system and nuclear reactor; In fact delays are caused by many phenomena like the time required to transport mass, energy or information, the time processing for sensors, the time needed for controllers to execute a complicated algorithm control and the accumulation of time lags in a number of simple plants connected in series.9,15 Delays have a considerable in°uence on the behavior of closed-loop system and can generate oscillations and even instability.5 PID controllers are of high interest thanks to their broad use in industrial circles.4 An analytical approach was developed in Refs. 1113 and allowed the characterization of the stability region of delayed systems controlled via PI and PID. Indeed, by using *This paper was recommended { Corresponding author.
by Regional Editor Piero Malcovati.
479
480
R. Farkh, K. Laabidii & M. Ksouri
the HermiteBiehler theorem applicable to the quasi-polynomials,2,13 the authors have developed an analytical characterization of all values of ðKp ; Ki ; Kd Þ for the case of ¯rst-order delay system. The same technique is used to provide a complete characterization of all P and PI controllers that stabilize a ¯rst-order delay system which is considered as less complicated than the PID stabilization problem.10 In Ref. 23, the characterization of the set of all stabilizing P/PI/PID parameters is given by using the HermiteBiehler Theorem for polynomials for a class of time delay system which veri¯es the interlacing property at high frequencies. Wang24 solved the problem of determining the PID stabilizing parameters set for high-order all pole time delay systems. In Ref. 26, Mahmoud presents a new delay-dependent stabilization scheme for a class of linear time-delay systems based on PID H1 feedback controller. These methods can be applied to synthesis of robust stabilizer controller for only interval plant without delay. The robustness is de¯ned as the performance and stability of plants that are exposed to uncertainties. Many important results in the robust control of system with parametric uncertainty have been based on the Kharitonov theorem.8 An important extension of Kharitonov's theorem is the edge theorem. The edge theorem states that the stability of a polytope of polynomials can be guaranteed by the stability of its one-dimensional exposed edge polynomials. Kharitonov's theorem has been generalized for the control problem which gives a necessary and su±cient condition for stabilization of interval plants; the advantage of the generalized Kharitonov theorem over the edge theorem is that the number of edges which are required for stability is not dependent on the number of uncertain parameters. The GKT reveals that a controller robustly stabilizes the system if it stabilizes a prescribed set of line segments in the plant parameter space.2,3 To determine the robust stability of time-delay system subjected to parametric uncertainty, the GKT and the edge theorem have been extended in the case of quasi-polynomials;2,17,18 a graphical approach based on the frequency response plots and on the zero exclusion principle has used too.20 In Ref. 19, the stability of a class of polytopic families of quasi-polynomials generated by multivariate interval polynomials is reduced to study the stability of a ¯nite number of vertex quasi-polynomials of the family. Mondi et al. presents an approach based on extensions of the ¯nite Nyquist theorem and the ¯nite inclusions theorem to quasi-polynomials to reduce the stability analysis of a polytopic family of quasipolynomials to the evaluation of a ¯nite number of quasi-polynomials at a ¯nite number of frequencies.22 A reduced stability-testing set for a diamond-like quasipolynomial family is presented in Ref. 21. There are some important results about stabilization of interval systems. Barmish et al. proved that a ¯rst-order controller stabilizes an interval plant if and only if it simultaneously stabilizes the sixteen Kharitonov plants family.1 Huang et al.7 used a parameter plan based on the gain phase margin tester method and the Kharitonov's theorem to obtain a nonconstructive region such that a PID controller stabilizes the entire interval plants. In Ref. 14, a proposed method based on the stability boundary locus was further used to
Robust Control for Uncertain Delay System
481
¯nd the stabilizing region of PI parameters for the control of a plant with uncertain parameters. Patre et al. proposed a two-degrees-of-freedom design methodology for interval process plants to guarantee both robust stability and performance.16 Based on the plant model in time domain, and by using the extraordinary feature results from the Kronecker sum operation, an explicit equation of control parameters de¯ning the stability boundary in parametric space is obtained. This method is proposed to compute all stabilizing PI and PID controllers for an uncertain system.25 In Refs. 6 and 13, the HermiteBiehler theorem was used for the formulation of the P, PI, PID controller to stabilize a delay-free interval plants family. In Ref. 27, the authors present some tools for designing PI and PID controllers for ¯rst-order systems with an unknowing delay with upper bound. In this paper, we propose to compute all PI gains which stabilize a ¯rst-order delay system where the coe±cients and the delay are subject to perturbation within prescribed ranges. We propose an approach based on combining the background presented in Sec. 4 and the result developed in Ref. 27. The validation of these results was tested in real time temperature control. This paper is organized as follows: the problem formulation are given in Sec. 2. Sections 3 and 4 are devoted for the robust stabilization problem for interval ¯rstorder system with uncertain time-delay controlled via PI controller. Section 5 is reserved for real process control to test the proposed approach.
2. Problem Formulation In this paper we propose a tool to design a robust PI controller for an uncertain ¯rstorder system with time-delay. We consider the plant family: GðsÞ ¼
Ke Ls ; 1 þ Ts
where K 2 ½K ; K , T 2 ½T ; T and L 2 ½L; L. By combined earlier robust stabilization results,27 which is presented in Sec. 3, and the approach developed in Sec. 4, we propose to design a robust PI controller that stabilizes the plant family GðsÞ. We demonstrate that to compute a robust stability region in parametric space of the PI controller, it is enough to stabilize a plant family where K 2 ½K ; K , T 2 ½T ; T and L ¼ L ¼ Lmax . 3. Robust Controller Design for Plant with Uncertain Time-Delay In this section we present the problem of stabilizing a ¯rst-order system with timedelay, where the parameters K and T are knowing and the time-delay is unknown but lies inside a known interval.
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R. Farkh, K. Laabidii & M. Ksouri
We consider the plant family: G1 ðsÞ ¼
Ke Ls ; 1 þ Ts
where L 2 ½L; L, K and T are known. Lemma 1. Consider the system with transfer function G1 ðsÞ. If a given PI controller stabilizes the delay-free system and the system with L ¼ L > 0 then the same PI controller stabilizes the system 8 L 2 ½0; L. Ls
Example 1. Consider the plant family G1 ðsÞ ¼ 1e1þs where L 2 ½1; 3 seconds. By using the algorithm presented in Ref. 11 we can ¯nd the set of stabilizing PI controllers for di®erent values of the time-delay. Figure 1 represents these sets for L ¼ 1; 1:5; 2; 3. The intersection of all these sets is the set corresponding to L ¼ 3 (dashed area). Thus, any PI controller from this set will stabilize the entire family of plants described by G1 ðsÞ. In view of Lemma 1, since the closed-loop system is stable for L, hence, it is stable for L 2 ½L; L. We can use this result to simplify the problem stabilization of the family plant GðsÞ to design a robust controller such that the following parameters veri¯es K 2 ½K ; K , T 2 ½T ; T and L ¼ L. In the rest of the paper the delay is set to L the known upper bound of the time-delay.
2 1.8 1.6
L=1
1.4
Ki
1.2
L=1.5
1 L=2 0.8 L=3
0.6 0.4 0.2 0 −1
−0.5
0
0.5
1
1.5
2
Kp Fig. 1. Sets of stabilizing PI controllers for Example 1.
2.5
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4. Robust Controller Design for Interval Plant with Time-Lag We focus on the problem of robust stabilization of delay system which belongs to the linear interval plant where the time-delay is a known constant. GðsÞ ¼
P1 ðsÞ Ls e ; P2 ðsÞ
ð1Þ
where P1 ðsÞ, P2 ðsÞ are linear interval polynomials. The problem studied in this paper can be formulated as follows: for a continuous time-delay system described by Eq. (1), the robust controller CðsÞ ¼ FF12 ðsÞ ðsÞ is to be designed with ¯xed polynomials F1 ðsÞ, F2 ðsÞ, such that the closed loop's robust stability is guaranteed. This problem can be solved by using the Generalized Kharitonov Theorem2 in which a ¯xed quasipolynomial (controller) robust stabilizes an interval polynomial (interval plant) where each coe±cient has bounded values. In fact, for mathematical analysis the delay location in feedback scheme is not a problem, therfore by making some changes in the scheme control, the delay will be associated with the controller as shown in Fig. 2. Indeed, Fig. 2(a) presents a feedback control where the plant contains a delay (thin dotted rectangle). This delay is located in their forward loop before it can be included with the controller as shown by Fig. 2(b) (thick dotted rectangle). Moreover, the control scheme consists of using a delay controller and a plant GðsÞ which will be seen as a delay-free interval system. Then, the transfer function and the F1 ðsÞ Ls controller can be expressed as GðsÞ ¼ PP12 ðsÞ . ðsÞ and CðsÞ ¼ F2 ðsÞ e
(a)
(b) Fig. 2. (a) Feedback control of time-delay system, (b) Feedback control using delay controller.
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R. Farkh, K. Laabidii & M. Ksouri
Hence, we are allowed to use the Generalized Kharitonov Theorem extended for quasi-polynomials to compute all stabilizing controller parameters for interval systems with time-delay. We recall some results from the parametric robust control area before stating the Generalized Kharitonov Theorem. Consider the family of quasi-polynomials ðsÞ: ðsÞ ¼ P1 ðsÞF1 ðsÞ þ P2 ðsÞF2 ðsÞ ;
ð2Þ
where .
P ðsÞ ¼ ðP1 ðsÞ; P2 ðsÞÞ is a ¯xed 2-tuple of real interval polynomials where each Pi ðsÞ is a real independent interval polynomial described as: Pi ðsÞ ¼ p0;i þ p1;i s þ þ pni ;i s ni ;
i ¼ 1; 2 :
ð3Þ
Pi ðsÞ is a linear interval polynomial characterized by the intervals Pj;i as: pj;i 2 ½p j;i ; p j;i ; .
i ¼ 1; 2 ;
j ¼ 0; 1; . . . ; ni :
ð4Þ
F ðsÞ ¼ ðF1 ðsÞ; F2 ðsÞÞ is a ¯xed 2-tuple of complex quasi-polynomials in this form: Fi ðsÞ ¼ F i0 ðsÞ þ F i1 ðsÞe sL i þ F i2 ðsÞe sL i þ : 1
2
ð5Þ
With the F ij ðsÞ being complex polynomials satisfying the following condition: degree½F i0 ðsÞ > degree½F ij ðsÞ ;
j 6¼ 0 :
ð6Þ
In our case, Fi ðsÞ with single delay is used as: Fi ðsÞ ¼ F i0 ðsÞ þ F i1 ðsÞe sLi . According to Ref. 2, the stability problem of Eq. (2) can be solved using the Generalized Kharitonov Theorem by constructing an extremal set of line segments E ðsÞ ðsÞ where the stability of E ðsÞ implies the stability of ðsÞ. E ðsÞ will be generated by constructing an extremal subset PE ðsÞ which is constructed from the Kharitonov polynomials of Pi ðsÞ. Theorem 1. Let F ¼ ðF1 ðsÞ; F2 ðsÞÞ a given 2-tuple of complex quasi-polynomials satisfying the condition (6) above and P ¼ ðP1 ðsÞ; P2 ðsÞÞ is an independent real interval polynomial: F ðsÞ stabilizes the entire family P ðsÞ if and only if F stabilize every 2-tuple segment in PE ðsÞ. Equivalently, ðsÞ is stable if and only if E ðsÞ is stable.2 Corollary 1. F ðsÞ stabilizes the linear system P ðsÞ if and only if the controller stabilizes the extremal transfer function GE ðsÞ ¼ PE ðsÞ, which will be detailed later. To use the GKT, the extremal set of line segment E ðsÞ must be determined ¯rst. From the segment polynomials of P1 ðsÞ and P2 ðsÞ, eight Kharitonov vertex equations are obtained and expressed by2,8: K 1m ðsÞ ; K 2m ðsÞ ;
m ¼ 1; 2; 3; 4 m ¼ 1; 2; 3; 4
for P1 ðsÞ ; for P2 ðsÞ ;
Robust Control for Uncertain Delay System
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where K i1 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ ; K i2 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ ; K i3 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ ;
ð7Þ
K i4 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ : The extremal subset P Ei ðsÞ; i ¼ 1; 2; consists of:3 P E1 ðsÞ ¼
K 1l ðsÞ þ ð1 ÞK 1k ðsÞ ; K 2h ðsÞ
K 1h ðsÞ P E2 ðsÞ ¼ ; l K 2 ðsÞ þ ð1 ÞK 2k ðsÞ
ð8Þ
where 2 ½0; 1, h ¼ 1; 2; 3; 4 and ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4. In the above equation, the number of extremal equations is ði4 i Þ where i presents the number of perturbed polynomials and ½l; k indicates connection points to make Kharitonov polytope K il ðsÞ þ ð1 ÞK ik ðsÞ; Some of the subset equations may be the same; hence, the extremal subset is described as:2 PE ðsÞ ¼ P E1 ðsÞ [ P E2 ðsÞ :
ð9Þ
The extremal subset of line segment (or generalized Kharitonov segment polynomials) is2: E ðsÞ ¼ 1E ðsÞ [ 2E ðsÞ ¼ fhF ðsÞ; P ðsÞi : P ðsÞ 2 PE ðsÞg ;
ð10Þ
hF ðsÞ; P ðsÞi ¼ F1 ðsÞP1 ðsÞ þ F2 ðsÞP2 ðsÞ :
ð11Þ
where
Knowing that E ðsÞ ðsÞ; if all polynomials of the linear interval system are stable, the system with perturbed parameters will be stable. On the following, the previous results on robust parametric approach control, which proved to be an e±cient control design technique, will be used for synthesis controllers that simultaneously stabilize a given interval plant with time-delay. 5. Robust PI Stabilization for Uncertain Time-Delay System In this section, we consider the problem of characterizing all PI controllers that Ls stabilize a given ¯rst-order interval plant with time-delay GðsÞ ¼ Ke 1þTs where K 2 ½K ; K , T 2 ½T ; T and L 2 ½L; L. By using the results developed in Secs. 3 and 4, the stabilization problem of the family plant GðsÞ is reduced to stabilize GðsÞ where K 2 ½K ; K , T 2 ½T ; T and L ¼ L.
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R. Farkh, K. Laabidii & M. Ksouri
The controller is given by: CðsÞ ¼ ðKp þ Ksi Þ. To obtain all PI gains that stabilize GðsÞ using the GKT for quasi-polynomials, K we take the new transfer function GðsÞ as: GðsÞ ¼ PP12 ðsÞ ðsÞ ¼ 1þTs and the new comF1 ðsÞ Ls Ki Ls pensator: CðsÞ ¼ F2 ðsÞ e ¼ ðKp þ s Þe . The family of closed loop characteristic quasi-polynomials ðs; Kp ; Ki Þ becomes: ðs; Kp ; Ki Þ ¼ P1 ðsÞF1 ðsÞ þ P2 ðsÞF2 ðsÞ ¼ KðKi þ Kp sÞe Ls þ ð1 þ TsÞs :
ð12Þ
The problem of characterizing all stabilizing PI controllers is to determine all the values of Kp and Ki for which the entire family of closed loop characteristic quasipolynomials is Hurwitz. Let K 1m ðsÞ; m ¼ 1; 2; 3; 4 and K 2m ðsÞ; m ¼ 1; 2; 3; 4 the Kharitonov polynomials corresponding to P1 ðsÞ ¼ K and P2 ðsÞ ¼ 1 þ Ts. K 11 ðsÞ ¼ K 12 ðsÞ ¼ K ; K 13 ðsÞ ¼ K 14 ðsÞ ¼ K ; K 21 ðsÞ ¼ K 23 ðsÞ ¼ 1 þ T s ; K 22 ðsÞ ¼ K 24 ðsÞ ¼ 1 þ T s : Let GE ðs; Þ denotes the family of 32 plant segments: 9 8 Glkh ðs; Þ= > > > > > > > > l k > > K ðsÞ þ ð1 ÞK ðsÞ > > 1 1 > > > > ðs; Þ ¼ G > > lkh h > > ðsÞ K > > 2 > > > > > > = < Or : GE ðs; Þ ¼ > > K 1h ðsÞ > > > > > ; > Glkh ðs; Þ ¼ > > > > K 2l ðsÞ þ ð1 ÞK 2k ðsÞ > > > > > > > > > > > > 2 ½0; 1; ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4; > > > > ; : h ¼ 1; 2; 3; 4
ð13Þ
Then, GE ðs; Þ consists of the following plant segments where the 32 extermal plants in Eq. (13) are reduced to 10. 9 8 1 1 þ ðK K Þ 1 1 þ ðK K Þ > > > > > > ; ; ; > > > > 1 þ 1 þ T s T s T s T s 1 þ 1 þ > > > > > > > > > > > > Or > > > > > > = < K K K K : ð14Þ GE ðs; Þ ¼ ; ; ; ; > > 1þTs 1þTs 1þTs > >1þTs > > > > > > > > K K > > > > ; > > > > > > 1 þ sðT 1 þ sðT þ ðT T ÞÞ þ ðT T ÞÞ > > > > > > ; : 2 ½0; 1
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The closed loop characteristic quasi-polynomial for each of these 10 plant segments Glkh ðs; Þ is denoted by lkh ðs; Kp ; Ki ; Þ and is de¯ned as: lkh ðs; Þ ¼ sDenðGlkh ðs; ÞÞ þ ðKi þ Kp sÞNumðGlkh ðs; ÞÞ ; where
8 l k h > : 2 ½0; 1 ; ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4 ; h ¼ 1; 2; 3; 4 :
We state the following theorem on stabilizing an interval ¯rst-order plant with time-delay using PI controller. Theorem 2. Let GðsÞ be a ¯rst-order interval plant with time-delay, the entire family GðsÞ is stabilized by a PI controller if and only if each Glkh ðs; Þ 2 GE ðs; Þ is stabilized by the same PI controller. Proof. Using Theorem 1, the entire family ðs; Kp ; Ki Þ is stable if and only if lkh ðs; Kp ; Ki ; Þ are all stable. Therefore, the entire family GðsÞ is stabilized by a PI controller if and only if every element of GE ðsÞ is simultaneously stabilized by the same PI. To get a characterization of all PI controllers that stabilize the interval plant GðsÞ by applying this procedure for each Glkh ðs; Þ belongs to GE ðs; Þ, the results in Refs. 11 and 13 will be used. 6. Experiments This section presents an application example for the temperature control of an air stream heater (process trainer P T 326), see Fig. 3. This type of process is found in many industrial systems such as furnaces, air conditioning, etc.
Fig. 3. PT-326 heat process trainer.
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R. Farkh, K. Laabidii & M. Ksouri
This process can be characterized as a linear time-delay system. Time-delay depends on the position of the temperature sensor element that can be inserted into the air stream at any one of the three points throughout the tube, spaced at 28, 140 and 280 mm from both the heater and the damper position. The system input, uðtÞ, is the voltage applied to the power circuit feeding the heating resistance, and the output, yðtÞ, is the outlet air temperature, expressed by a voltage, between 10 V and 10 V. The P T 326 heating process is shown in Fig. 4. The behavior of the P T 326 thermal process is governed by the balance of heat energy. When the air temperature inside the tube is supposed to be uniform, a linear delay system model can be obtained. Thus, the transfer function between the heater input voltage and the sensor output voltage can be obtained as: yðsÞ K ¼ e Ls : uðsÞ 1 þ Ts For the experiment, the damper position is set to 30, and the temperature sensor is placed in the third position. The measurements acquisition is done by the \P CI DAS1002" card's, the sampling time is taken equal to Te ¼ 0:03 second. This choice takes account of the time computing and the constant time of the plant. Firstly data is taken by operating the heater open loop with square input 0 V=2 V as shown as Fig. 5. The data is then used to obtain models to represent the plant. An enlargement of the ¯rst step response up to 1500 iterations is shown in Fig. 6. We de¯ne: Step 1: Step response up to 1500 iterations Step 3: Step response from 3000 up to 4500 iterations Heater
I
II
III
Air output 30°
Air input Power Supply
Bridge Circuit
u(t)
y(t)
Fig. 4.
Schematic illustration.
Robust Control for Uncertain Delay System
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2.5 y(k) u(k) 2
u(k), y(k) [V]
1.5
1
0.5
0
−0.5
0
2000
4000
6000 itérations (k)
8000
10000
12000
Fig. 5. Square input 0 V=2 V and outputs of P T 326. 2.5 y(k) u(k) 2
u(k), y(k) [V]
1.5
1
0.5
0
−0.5
0
500
1000
1500
itérations (k)
Fig. 6.
First step response for PT 326.
Step 5: Step response from 6000 up to 7500 iterations Step 7: Step response from 9000 up to 10500 iterations The transfer's functions corresponding to each step response are estimated by the System Identi¯cation Toolbox of Matlab which is based on recursive estimation algorithms. The estimated parameters are represented in Table 1.
490
R. Farkh, K. Laabidii & M. Ksouri Table 1. System parameters.
L K T
Step 1
Step 3
Step 5
Step 7
0.57 0.5874 1.6201
0.6 0.6256 1.5671
0.54 0.6597 1.922
0.57 0.6825 1.7265
According to this table, that the coe±cient in the following bounds 0:58 < K < 0:68, 1:56 < T < 1:92 and 0:54 < L < 0:6. Ls We consider the plant family GðsÞ ¼ Ke 1þTs where 0:58 < K < 0:68, 1:56 < T < 1:92 and 0:54 < L < 0:6. The ¯nal PI stability region that guarantee stability of GðsÞ, under parameter varying case, is included in that one obtained by using only the maximum delay L ¼ L ¼ 0:6. The entire family GE ðs; Þ is given as follows: 9 8 Gij ðs; Þ= > > > > > > > > > > l k > l K 1 ðsÞ þ ð1 l ÞK 1 ðsÞ > > > > > > > G ðs; Þ ¼ ij > > h ðsÞ > > K > > 2 > > > > S > > = < GE ðs; Þ ¼ > > K 1h ðsÞ > > > > > > ðs; Þ ¼ G ji > > l k > > K ðsÞ þ ð1 ÞK ðsÞ > > m m 2 2 > > > > > > > > > > > > 2 ½0; 1; h ¼ 1; 2; 3; 4; > > > > ; : ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4 According to Eq. (14), we obtain: 9 8 Gij ðs; Þ= > > > > > > > > > > 1 1 > > > > ; g ; g ¼ ¼ > > 1 2 > > 1 þ 1:56s 1 þ 1:92s > > > > > > > > > > > > 1 þ 0:1 1 þ 0:1 > > > > ; g ; g ¼ ¼ > > 3 4 > > > > 1 þ 1:56s 1 þ 1:92s > > > > > > S > > > > > > = < 0:58 0:68 GE ðs; Þ ¼ ; g6 ¼ ; g ¼ > > > > > > 5 1 þ 1:56s 1 þ 1:56s > > > > > > > > > > 0:58 0:68 > > > > g ; g ; ¼ ¼ > > 7 8 > > > 1 þ 1:92s 1 þ sð1:56 þ 0:36Þ > > > > > > > > > > > 0:58 0:68 > > > g9 ¼ ; g10 ¼ ;> > > > > > > 1 þ sð1:56 þ 0:36Þ 1 þ 1:92s > > > > > > ; : 2 ½0; 1
Robust Control for Uncertain Delay System
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We remark that g3 , g4 , g8 and g9 have an in¯nity of transfer function's sets due to their dependence in . To reduce the complexity of this problem, we set at 0, 0:2, 0:5, 0:7 and 1 as di®erent values of 2 ½0; 1. We de¯ne ghp ¼ gh ðs; p Þ where p 2 f0; 0:2; 0:5; 0:7; 1g for h ¼ 1; . . . ; 10; and p ¼ 1; . . . ; 5. So we obtain: 8 gh1 ¼ gh ðs; ¼ 0Þ ; > > > > > > > g ¼ gh ðs; ¼ 0:7Þ ; > > : h4 gh5 ¼ gh ðs; ¼ 1Þ : Figure 7 presents the stabilizing ðKp ; Ki Þ values for the entire family G by invoking the result presented in Ref. 11 which is applying for each transfer function that belongs to GE . The intersection of these stabilities regions presents an overlapped area of the boundaries which constitutes the entire feasible controller sets that stabilize the entire family GðsÞ. We choose some PI controller parameters inside the stability region given by Fig. 8 and which are presented in Table 2.
9 8 7 6
Ki
5 4 3 2 1 −2
0
2
4
6
8
10
Kp Fig. 7. The stabilizing set of ðKp ; Ki Þ for GðsÞ.
g1 g2 g3 g32 g33 g34 g35 g4 g42 g43 g44 g5 g6 g7 g82 g83 g84 g92 g93 g94 g10
492
R. Farkh, K. Laabidii & M. Ksouri
5 g3 4.5 4 3.5
Ki
3 2.5 2 1.5 1 0.5 0 −1
0
1
2 Kp
3
4
5
Fig. 8. Final stability region in ðKp ; Ki Þ plan for uncertain plant GðsÞ.
Table 2. PI controllers. Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Kp
0.1
0.9
1
1
1.37
1.54
2.94
3.33
3.67
3.95
Ki
0.1
0.7
0.33
0.65
2.77
1.18
4.04
4.36
4.34
4.33
The following algorithm describes the real-time implementation of the PI control law: (1) (2) (3) (4) (5) (6)
Initialization k ¼ 1, Output acquisition yðkÞ, Error computation eðkÞ ¼ yc ðkÞ yðkÞ, Control law computation uðkÞ ¼ uðk 1Þ þ Kp eðkÞ þ ðKi Te Kp Þeðk 1Þ, Application of the control law uðkÞ to the process Exhausting of the sampling time, k ¼ k þ 1 then go to Step 2.
The systems outputs and control signals respectively presented in Figs. 918 in the case of using di®erent PI controllers as described in Table 2.
Robust Control for Uncertain Delay System
7
6
u(k),y(k) [V]
5
4
3
2
1
0
y(k) u(k) 0
1000
2000
3000 iterations
4000
5000
6000
Fig. 9. Evolution of the output and the control law (case 1).
7
6
u(k),y(k) [V]
5
4
3
2 y(k) u(k)
1
0
0
1000
2000
3000 iterations
4000
5000
Fig. 10. Evolution of the output and the control law (case 2).
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7
6
y(k),u(k) [V]
5
4
3
2 u(k) y(k)
1
0 0
1000
2000
3000 iterations
4000
5000
6000
Fig. 11. Evolution of the output and the control law (case 3).
6.5 6 5.5 5 u(k),y(k) [V]
494
4.5 4 3.5 3 y(k) u(k)
2.5 2 1.5
0
1000
2000
3000 iterations
4000
5000
Fig. 12. Evolution of the output and the control law (case 4).
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Robust Control for Uncertain Delay System
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6
y(k),u(k) [V]
5
4
3
2
1
0
y(k) u(k) 0
1000
2000
3000 iterations
4000
5000
6000
Fig. 13. Evolution of the output and the control law (case 5).
7
6
u(k),y(k) [V]
5
4
3
2 y(k) u(k)
1
0
0
1000
2000
3000 iterations
4000
5000
Fig. 14. Evolution of the output and the control law (case 6).
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8 7 6
u(k),y(k) [V]
5 4 3 2 1 u(k) y(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
6000
Fig. 15. Evolution of the output and the control law (case 7).
8 7 6 5 u(k),y(k) [V]
496
4 3 2 1
y(k) u(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
Fig. 16. Evolution of the output and the control law (case 8).
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Robust Control for Uncertain Delay System
8 7 6
u(k),y(k) [V]
5 4 3 2 1
u(k) y(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
6000
Fig. 17. Evolution of the output and the control law (case 9).
8 7 6
u(k),y(k) [V]
5 4 3 2 1 y(k) u(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
Fig. 18. Evolution of the output and the control law (case 10).
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7. Conclusion In this paper, a version of the HermiteBiehler theorem and Generalized Kharitonov Theorem is adopted and proposed to ¯nd the PI stability region for the control of an uncertain ¯rst-order plant with time-delay. This stabilization problem is reduced to the equivalent problem of stabilizing a ¯nite set of 10 external plants. Also, an air stream heater has been controlled using a robust PI controller to demonstrate the e®ectiveness of proposed technique.
References 1. B. R. Barmish, C. V. Holot, F. J. Kraus and R. Tempo, Extreme points results for robust stabilization of interval plants with ¯rst-order compensators, IEEE Trans. Automat. Contr. 38 (1993) 17341735. 2. S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach (Prentice-Hall, Upper Saddle River, 1995). 3. S. P. Bhattacharyya and H. Chapellat, A generalization of Kharitonov's Theorem: Robust stability interval plants, IEEE Trans. Automat. Contr. AC-34 (1989) 306311. 4. C. Knospe and G. Editor, PID control, IEEE Contr. Syst. Mag. (2006). 5. L. Dugard and E. I. Verriest, Stability and Control for Time-Delay System (Springer, 1998). 6. M. T. Ho, A. Datta and S. P. Bhattacharyya, Design of P, PI and PID controllers for interval plants, Proc. American Control Conf. (1998). 7. Y. J. Huang and Y. J. Wang, Robust PID tuning strategy for uncertain plants based on the Kharitonov theorem, ISA Trans. 39 (2000) 419431. 8. V. L. Kharitonov, Asymptotic stability of an equilibrium position of a family of systems of linear di®erential equations, Di®er. Equat. 14 (1979) 148. 9. S.-I. Niculescu, Delay E®ects on Stability (Springer, London, 2001). 10. G. J. Silva, A. Datta and S. P. Bhattacharyya, Stabilization of time-delay systems, Proc. American Control Conf. (2000), pp. 963970. 11. G. J. Silva, A. Datta and S. P. Bhattacharyya, PI stabilization of ¯rst-order systems with time-delay, Automatica 37 (2001) 20252031. 12. G. J. Silva, A. Datta and S. P. Bhattacharyya, Stabilization of ¯rst-order systems with time-delay using the PID controller, Proc. American Control Conf. (2001), pp. 46504655. 13. G. J. Silva, A. Datta and S. P. Bhattacharyya, PID Controllers for Time-Delay Systems (Springer, London, 2005). 14. N. Tan, I. Kaya, C. Yeroglu and D. P. Atherton, Computation of stabilizing PI and PID controllers using the stability boundary locus, Energ. Convers. Manag. 47 (2006) 30453058. 15. Q. C. Zhong, Robust Control of Time-Delay System (Springer, London, 2006). 16. B. M. Patre and P. J. Deore, Robust stability and performance for interval process plants, ISA Tran. 46 (2007) 343349. 17. K. Gu, V. Kharitonov and J. Chen, Stability of Time-Delay Systems (Birkhäuser, Boston, 2003). 18. K. Gu and V. Kharitonov, Robust stability of time-delay systems, IEEE Trans. Automat. Contr. 39 (1994).
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19. V. L. Kharitonov, J. A. Torres-Muñoz and M. B. Ortiz-Moctezuma, Polytopic families of quasi-polynomials: Vertex-type stability conditions, IEEE Trans. Circuit Syst. 50 (2003). 20. M. Fu, A. W. Olbrot and M. P. Polis, Robust stability for time-delay systems: The edge theorem and graphical tests, IEEE Trans. Automat. Contr. 34 (1989) 813819. 21. V. L. Kharitonov, J. A. Torres-Muñoz and M. B. Ortiz-Moctezuma, Stability of a multidiamond type family of quasi-polynomials, Proc. 47th IEEE Conf. on Decision and Control. 22. S. Mondie, J. Santos and V. L. Kharitonov, Robust stability of quasi-polynomials and the ¯nite inclusions theorem, IEEE Trans. Automat. Contr. 50 (2005). 23. V. A. Oliveira, L. V. Cossi, C. M. Teixeira and A. M. F. Silva, Synthesis of PID controllers for a class of time-delay systems, Automatica, doi:10.1016/j.automatica.2009.03.018. 24. D. J. Wang, Synthesis of PID controllers for high-order plants with time-delay, J. Process Control (2009). 25. J. Fang, D. Zheng and Z. Ren, Computation of stabilizing PI and PID controllers by using Kronecker summation method, Energ. Conver. Manag. 50 (2009) 18211827. 26. M. S. Mahmoud, Stabilization of time-delay systems by PID controllers, Int. Multi-Conf. Systems, Signals and Devices, SSD'09. 27. G. J. Silva, A. Datta and S. P. Bhattacharyya, Robust control design using PID controller, IEEE Conf. Decision and Control (2002).
ROBUST CONTROL FOR UNCERTAIN DELAY SYSTEM
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RIHEM FARKHy,{, KAOUTHER LAABIDIIz and MEKKI KSOURIx Research unit on Systems Analysis and Control, National School Engineering of Tunis, BP, 37, Le Belvedere, 1002 Tunis, Tunisia y [email protected] [email protected] [email protected] Received 6 February 2010 Accepted 30 November 2010 In this paper, the Generalized Kharitonov Theorem for quasi-polynomials is exploited for the purpose of synthesizing a robust controller. The aim here is to develop a controller to simultaneously stabilize a given interval plant family with unknowing and bounded time delay. Using a constructive procedure based on HermiteBiehler theorem, we obtain all PI gains that stabilize an uncertain ¯rst-order delay system. An application example is presented for the temperature control of a heated air stream, process trainer P T 326. Keywords: Uncertain time delay system; interval plants; Generalized Kharitonov Theorem (GKT); PI controller; HermiteBiehler theorem; stability region.
1. Introduction Dead times are often encountered in various engineering systems and industry processes such as electrical and communication network, turbojet engine, chemical process, hydraulic system and nuclear reactor; In fact delays are caused by many phenomena like the time required to transport mass, energy or information, the time processing for sensors, the time needed for controllers to execute a complicated algorithm control and the accumulation of time lags in a number of simple plants connected in series.9,15 Delays have a considerable in°uence on the behavior of closed-loop system and can generate oscillations and even instability.5 PID controllers are of high interest thanks to their broad use in industrial circles.4 An analytical approach was developed in Refs. 1113 and allowed the characterization of the stability region of delayed systems controlled via PI and PID. Indeed, by using *This paper was recommended { Corresponding author.
by Regional Editor Piero Malcovati.
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the HermiteBiehler theorem applicable to the quasi-polynomials,2,13 the authors have developed an analytical characterization of all values of ðKp ; Ki ; Kd Þ for the case of ¯rst-order delay system. The same technique is used to provide a complete characterization of all P and PI controllers that stabilize a ¯rst-order delay system which is considered as less complicated than the PID stabilization problem.10 In Ref. 23, the characterization of the set of all stabilizing P/PI/PID parameters is given by using the HermiteBiehler Theorem for polynomials for a class of time delay system which veri¯es the interlacing property at high frequencies. Wang24 solved the problem of determining the PID stabilizing parameters set for high-order all pole time delay systems. In Ref. 26, Mahmoud presents a new delay-dependent stabilization scheme for a class of linear time-delay systems based on PID H1 feedback controller. These methods can be applied to synthesis of robust stabilizer controller for only interval plant without delay. The robustness is de¯ned as the performance and stability of plants that are exposed to uncertainties. Many important results in the robust control of system with parametric uncertainty have been based on the Kharitonov theorem.8 An important extension of Kharitonov's theorem is the edge theorem. The edge theorem states that the stability of a polytope of polynomials can be guaranteed by the stability of its one-dimensional exposed edge polynomials. Kharitonov's theorem has been generalized for the control problem which gives a necessary and su±cient condition for stabilization of interval plants; the advantage of the generalized Kharitonov theorem over the edge theorem is that the number of edges which are required for stability is not dependent on the number of uncertain parameters. The GKT reveals that a controller robustly stabilizes the system if it stabilizes a prescribed set of line segments in the plant parameter space.2,3 To determine the robust stability of time-delay system subjected to parametric uncertainty, the GKT and the edge theorem have been extended in the case of quasi-polynomials;2,17,18 a graphical approach based on the frequency response plots and on the zero exclusion principle has used too.20 In Ref. 19, the stability of a class of polytopic families of quasi-polynomials generated by multivariate interval polynomials is reduced to study the stability of a ¯nite number of vertex quasi-polynomials of the family. Mondi et al. presents an approach based on extensions of the ¯nite Nyquist theorem and the ¯nite inclusions theorem to quasi-polynomials to reduce the stability analysis of a polytopic family of quasipolynomials to the evaluation of a ¯nite number of quasi-polynomials at a ¯nite number of frequencies.22 A reduced stability-testing set for a diamond-like quasipolynomial family is presented in Ref. 21. There are some important results about stabilization of interval systems. Barmish et al. proved that a ¯rst-order controller stabilizes an interval plant if and only if it simultaneously stabilizes the sixteen Kharitonov plants family.1 Huang et al.7 used a parameter plan based on the gain phase margin tester method and the Kharitonov's theorem to obtain a nonconstructive region such that a PID controller stabilizes the entire interval plants. In Ref. 14, a proposed method based on the stability boundary locus was further used to
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¯nd the stabilizing region of PI parameters for the control of a plant with uncertain parameters. Patre et al. proposed a two-degrees-of-freedom design methodology for interval process plants to guarantee both robust stability and performance.16 Based on the plant model in time domain, and by using the extraordinary feature results from the Kronecker sum operation, an explicit equation of control parameters de¯ning the stability boundary in parametric space is obtained. This method is proposed to compute all stabilizing PI and PID controllers for an uncertain system.25 In Refs. 6 and 13, the HermiteBiehler theorem was used for the formulation of the P, PI, PID controller to stabilize a delay-free interval plants family. In Ref. 27, the authors present some tools for designing PI and PID controllers for ¯rst-order systems with an unknowing delay with upper bound. In this paper, we propose to compute all PI gains which stabilize a ¯rst-order delay system where the coe±cients and the delay are subject to perturbation within prescribed ranges. We propose an approach based on combining the background presented in Sec. 4 and the result developed in Ref. 27. The validation of these results was tested in real time temperature control. This paper is organized as follows: the problem formulation are given in Sec. 2. Sections 3 and 4 are devoted for the robust stabilization problem for interval ¯rstorder system with uncertain time-delay controlled via PI controller. Section 5 is reserved for real process control to test the proposed approach.
2. Problem Formulation In this paper we propose a tool to design a robust PI controller for an uncertain ¯rstorder system with time-delay. We consider the plant family: GðsÞ ¼
Ke Ls ; 1 þ Ts
where K 2 ½K ; K , T 2 ½T ; T and L 2 ½L; L. By combined earlier robust stabilization results,27 which is presented in Sec. 3, and the approach developed in Sec. 4, we propose to design a robust PI controller that stabilizes the plant family GðsÞ. We demonstrate that to compute a robust stability region in parametric space of the PI controller, it is enough to stabilize a plant family where K 2 ½K ; K , T 2 ½T ; T and L ¼ L ¼ Lmax . 3. Robust Controller Design for Plant with Uncertain Time-Delay In this section we present the problem of stabilizing a ¯rst-order system with timedelay, where the parameters K and T are knowing and the time-delay is unknown but lies inside a known interval.
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We consider the plant family: G1 ðsÞ ¼
Ke Ls ; 1 þ Ts
where L 2 ½L; L, K and T are known. Lemma 1. Consider the system with transfer function G1 ðsÞ. If a given PI controller stabilizes the delay-free system and the system with L ¼ L > 0 then the same PI controller stabilizes the system 8 L 2 ½0; L. Ls
Example 1. Consider the plant family G1 ðsÞ ¼ 1e1þs where L 2 ½1; 3 seconds. By using the algorithm presented in Ref. 11 we can ¯nd the set of stabilizing PI controllers for di®erent values of the time-delay. Figure 1 represents these sets for L ¼ 1; 1:5; 2; 3. The intersection of all these sets is the set corresponding to L ¼ 3 (dashed area). Thus, any PI controller from this set will stabilize the entire family of plants described by G1 ðsÞ. In view of Lemma 1, since the closed-loop system is stable for L, hence, it is stable for L 2 ½L; L. We can use this result to simplify the problem stabilization of the family plant GðsÞ to design a robust controller such that the following parameters veri¯es K 2 ½K ; K , T 2 ½T ; T and L ¼ L. In the rest of the paper the delay is set to L the known upper bound of the time-delay.
2 1.8 1.6
L=1
1.4
Ki
1.2
L=1.5
1 L=2 0.8 L=3
0.6 0.4 0.2 0 −1
−0.5
0
0.5
1
1.5
2
Kp Fig. 1. Sets of stabilizing PI controllers for Example 1.
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4. Robust Controller Design for Interval Plant with Time-Lag We focus on the problem of robust stabilization of delay system which belongs to the linear interval plant where the time-delay is a known constant. GðsÞ ¼
P1 ðsÞ Ls e ; P2 ðsÞ
ð1Þ
where P1 ðsÞ, P2 ðsÞ are linear interval polynomials. The problem studied in this paper can be formulated as follows: for a continuous time-delay system described by Eq. (1), the robust controller CðsÞ ¼ FF12 ðsÞ ðsÞ is to be designed with ¯xed polynomials F1 ðsÞ, F2 ðsÞ, such that the closed loop's robust stability is guaranteed. This problem can be solved by using the Generalized Kharitonov Theorem2 in which a ¯xed quasipolynomial (controller) robust stabilizes an interval polynomial (interval plant) where each coe±cient has bounded values. In fact, for mathematical analysis the delay location in feedback scheme is not a problem, therfore by making some changes in the scheme control, the delay will be associated with the controller as shown in Fig. 2. Indeed, Fig. 2(a) presents a feedback control where the plant contains a delay (thin dotted rectangle). This delay is located in their forward loop before it can be included with the controller as shown by Fig. 2(b) (thick dotted rectangle). Moreover, the control scheme consists of using a delay controller and a plant GðsÞ which will be seen as a delay-free interval system. Then, the transfer function and the F1 ðsÞ Ls controller can be expressed as GðsÞ ¼ PP12 ðsÞ . ðsÞ and CðsÞ ¼ F2 ðsÞ e
(a)
(b) Fig. 2. (a) Feedback control of time-delay system, (b) Feedback control using delay controller.
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Hence, we are allowed to use the Generalized Kharitonov Theorem extended for quasi-polynomials to compute all stabilizing controller parameters for interval systems with time-delay. We recall some results from the parametric robust control area before stating the Generalized Kharitonov Theorem. Consider the family of quasi-polynomials ðsÞ: ðsÞ ¼ P1 ðsÞF1 ðsÞ þ P2 ðsÞF2 ðsÞ ;
ð2Þ
where .
P ðsÞ ¼ ðP1 ðsÞ; P2 ðsÞÞ is a ¯xed 2-tuple of real interval polynomials where each Pi ðsÞ is a real independent interval polynomial described as: Pi ðsÞ ¼ p0;i þ p1;i s þ þ pni ;i s ni ;
i ¼ 1; 2 :
ð3Þ
Pi ðsÞ is a linear interval polynomial characterized by the intervals Pj;i as: pj;i 2 ½p j;i ; p j;i ; .
i ¼ 1; 2 ;
j ¼ 0; 1; . . . ; ni :
ð4Þ
F ðsÞ ¼ ðF1 ðsÞ; F2 ðsÞÞ is a ¯xed 2-tuple of complex quasi-polynomials in this form: Fi ðsÞ ¼ F i0 ðsÞ þ F i1 ðsÞe sL i þ F i2 ðsÞe sL i þ : 1
2
ð5Þ
With the F ij ðsÞ being complex polynomials satisfying the following condition: degree½F i0 ðsÞ > degree½F ij ðsÞ ;
j 6¼ 0 :
ð6Þ
In our case, Fi ðsÞ with single delay is used as: Fi ðsÞ ¼ F i0 ðsÞ þ F i1 ðsÞe sLi . According to Ref. 2, the stability problem of Eq. (2) can be solved using the Generalized Kharitonov Theorem by constructing an extremal set of line segments E ðsÞ ðsÞ where the stability of E ðsÞ implies the stability of ðsÞ. E ðsÞ will be generated by constructing an extremal subset PE ðsÞ which is constructed from the Kharitonov polynomials of Pi ðsÞ. Theorem 1. Let F ¼ ðF1 ðsÞ; F2 ðsÞÞ a given 2-tuple of complex quasi-polynomials satisfying the condition (6) above and P ¼ ðP1 ðsÞ; P2 ðsÞÞ is an independent real interval polynomial: F ðsÞ stabilizes the entire family P ðsÞ if and only if F stabilize every 2-tuple segment in PE ðsÞ. Equivalently, ðsÞ is stable if and only if E ðsÞ is stable.2 Corollary 1. F ðsÞ stabilizes the linear system P ðsÞ if and only if the controller stabilizes the extremal transfer function GE ðsÞ ¼ PE ðsÞ, which will be detailed later. To use the GKT, the extremal set of line segment E ðsÞ must be determined ¯rst. From the segment polynomials of P1 ðsÞ and P2 ðsÞ, eight Kharitonov vertex equations are obtained and expressed by2,8: K 1m ðsÞ ; K 2m ðsÞ ;
m ¼ 1; 2; 3; 4 m ¼ 1; 2; 3; 4
for P1 ðsÞ ; for P2 ðsÞ ;
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where K i1 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ ; K i2 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ ; K i3 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ ;
ð7Þ
K i4 ðsÞ ¼ p i;0 þ p i;1 s 1 þ p i;2 s 2 þ p i;3 s 3 þ : The extremal subset P Ei ðsÞ; i ¼ 1; 2; consists of:3 P E1 ðsÞ ¼
K 1l ðsÞ þ ð1 ÞK 1k ðsÞ ; K 2h ðsÞ
K 1h ðsÞ P E2 ðsÞ ¼ ; l K 2 ðsÞ þ ð1 ÞK 2k ðsÞ
ð8Þ
where 2 ½0; 1, h ¼ 1; 2; 3; 4 and ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4. In the above equation, the number of extremal equations is ði4 i Þ where i presents the number of perturbed polynomials and ½l; k indicates connection points to make Kharitonov polytope K il ðsÞ þ ð1 ÞK ik ðsÞ; Some of the subset equations may be the same; hence, the extremal subset is described as:2 PE ðsÞ ¼ P E1 ðsÞ [ P E2 ðsÞ :
ð9Þ
The extremal subset of line segment (or generalized Kharitonov segment polynomials) is2: E ðsÞ ¼ 1E ðsÞ [ 2E ðsÞ ¼ fhF ðsÞ; P ðsÞi : P ðsÞ 2 PE ðsÞg ;
ð10Þ
hF ðsÞ; P ðsÞi ¼ F1 ðsÞP1 ðsÞ þ F2 ðsÞP2 ðsÞ :
ð11Þ
where
Knowing that E ðsÞ ðsÞ; if all polynomials of the linear interval system are stable, the system with perturbed parameters will be stable. On the following, the previous results on robust parametric approach control, which proved to be an e±cient control design technique, will be used for synthesis controllers that simultaneously stabilize a given interval plant with time-delay. 5. Robust PI Stabilization for Uncertain Time-Delay System In this section, we consider the problem of characterizing all PI controllers that Ls stabilize a given ¯rst-order interval plant with time-delay GðsÞ ¼ Ke 1þTs where K 2 ½K ; K , T 2 ½T ; T and L 2 ½L; L. By using the results developed in Secs. 3 and 4, the stabilization problem of the family plant GðsÞ is reduced to stabilize GðsÞ where K 2 ½K ; K , T 2 ½T ; T and L ¼ L.
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The controller is given by: CðsÞ ¼ ðKp þ Ksi Þ. To obtain all PI gains that stabilize GðsÞ using the GKT for quasi-polynomials, K we take the new transfer function GðsÞ as: GðsÞ ¼ PP12 ðsÞ ðsÞ ¼ 1þTs and the new comF1 ðsÞ Ls Ki Ls pensator: CðsÞ ¼ F2 ðsÞ e ¼ ðKp þ s Þe . The family of closed loop characteristic quasi-polynomials ðs; Kp ; Ki Þ becomes: ðs; Kp ; Ki Þ ¼ P1 ðsÞF1 ðsÞ þ P2 ðsÞF2 ðsÞ ¼ KðKi þ Kp sÞe Ls þ ð1 þ TsÞs :
ð12Þ
The problem of characterizing all stabilizing PI controllers is to determine all the values of Kp and Ki for which the entire family of closed loop characteristic quasipolynomials is Hurwitz. Let K 1m ðsÞ; m ¼ 1; 2; 3; 4 and K 2m ðsÞ; m ¼ 1; 2; 3; 4 the Kharitonov polynomials corresponding to P1 ðsÞ ¼ K and P2 ðsÞ ¼ 1 þ Ts. K 11 ðsÞ ¼ K 12 ðsÞ ¼ K ; K 13 ðsÞ ¼ K 14 ðsÞ ¼ K ; K 21 ðsÞ ¼ K 23 ðsÞ ¼ 1 þ T s ; K 22 ðsÞ ¼ K 24 ðsÞ ¼ 1 þ T s : Let GE ðs; Þ denotes the family of 32 plant segments: 9 8 Glkh ðs; Þ= > > > > > > > > l k > > K ðsÞ þ ð1 ÞK ðsÞ > > 1 1 > > > > ðs; Þ ¼ G > > lkh h > > ðsÞ K > > 2 > > > > > > = < Or : GE ðs; Þ ¼ > > K 1h ðsÞ > > > > > ; > Glkh ðs; Þ ¼ > > > > K 2l ðsÞ þ ð1 ÞK 2k ðsÞ > > > > > > > > > > > > 2 ½0; 1; ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4; > > > > ; : h ¼ 1; 2; 3; 4
ð13Þ
Then, GE ðs; Þ consists of the following plant segments where the 32 extermal plants in Eq. (13) are reduced to 10. 9 8 1 1 þ ðK K Þ 1 1 þ ðK K Þ > > > > > > ; ; ; > > > > 1 þ 1 þ T s T s T s T s 1 þ 1 þ > > > > > > > > > > > > Or > > > > > > = < K K K K : ð14Þ GE ðs; Þ ¼ ; ; ; ; > > 1þTs 1þTs 1þTs > >1þTs > > > > > > > > K K > > > > ; > > > > > > 1 þ sðT 1 þ sðT þ ðT T ÞÞ þ ðT T ÞÞ > > > > > > ; : 2 ½0; 1
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The closed loop characteristic quasi-polynomial for each of these 10 plant segments Glkh ðs; Þ is denoted by lkh ðs; Kp ; Ki ; Þ and is de¯ned as: lkh ðs; Þ ¼ sDenðGlkh ðs; ÞÞ þ ðKi þ Kp sÞNumðGlkh ðs; ÞÞ ; where
8 l k h > : 2 ½0; 1 ; ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4 ; h ¼ 1; 2; 3; 4 :
We state the following theorem on stabilizing an interval ¯rst-order plant with time-delay using PI controller. Theorem 2. Let GðsÞ be a ¯rst-order interval plant with time-delay, the entire family GðsÞ is stabilized by a PI controller if and only if each Glkh ðs; Þ 2 GE ðs; Þ is stabilized by the same PI controller. Proof. Using Theorem 1, the entire family ðs; Kp ; Ki Þ is stable if and only if lkh ðs; Kp ; Ki ; Þ are all stable. Therefore, the entire family GðsÞ is stabilized by a PI controller if and only if every element of GE ðsÞ is simultaneously stabilized by the same PI. To get a characterization of all PI controllers that stabilize the interval plant GðsÞ by applying this procedure for each Glkh ðs; Þ belongs to GE ðs; Þ, the results in Refs. 11 and 13 will be used. 6. Experiments This section presents an application example for the temperature control of an air stream heater (process trainer P T 326), see Fig. 3. This type of process is found in many industrial systems such as furnaces, air conditioning, etc.
Fig. 3. PT-326 heat process trainer.
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This process can be characterized as a linear time-delay system. Time-delay depends on the position of the temperature sensor element that can be inserted into the air stream at any one of the three points throughout the tube, spaced at 28, 140 and 280 mm from both the heater and the damper position. The system input, uðtÞ, is the voltage applied to the power circuit feeding the heating resistance, and the output, yðtÞ, is the outlet air temperature, expressed by a voltage, between 10 V and 10 V. The P T 326 heating process is shown in Fig. 4. The behavior of the P T 326 thermal process is governed by the balance of heat energy. When the air temperature inside the tube is supposed to be uniform, a linear delay system model can be obtained. Thus, the transfer function between the heater input voltage and the sensor output voltage can be obtained as: yðsÞ K ¼ e Ls : uðsÞ 1 þ Ts For the experiment, the damper position is set to 30, and the temperature sensor is placed in the third position. The measurements acquisition is done by the \P CI DAS1002" card's, the sampling time is taken equal to Te ¼ 0:03 second. This choice takes account of the time computing and the constant time of the plant. Firstly data is taken by operating the heater open loop with square input 0 V=2 V as shown as Fig. 5. The data is then used to obtain models to represent the plant. An enlargement of the ¯rst step response up to 1500 iterations is shown in Fig. 6. We de¯ne: Step 1: Step response up to 1500 iterations Step 3: Step response from 3000 up to 4500 iterations Heater
I
II
III
Air output 30°
Air input Power Supply
Bridge Circuit
u(t)
y(t)
Fig. 4.
Schematic illustration.
Robust Control for Uncertain Delay System
489
2.5 y(k) u(k) 2
u(k), y(k) [V]
1.5
1
0.5
0
−0.5
0
2000
4000
6000 itérations (k)
8000
10000
12000
Fig. 5. Square input 0 V=2 V and outputs of P T 326. 2.5 y(k) u(k) 2
u(k), y(k) [V]
1.5
1
0.5
0
−0.5
0
500
1000
1500
itérations (k)
Fig. 6.
First step response for PT 326.
Step 5: Step response from 6000 up to 7500 iterations Step 7: Step response from 9000 up to 10500 iterations The transfer's functions corresponding to each step response are estimated by the System Identi¯cation Toolbox of Matlab which is based on recursive estimation algorithms. The estimated parameters are represented in Table 1.
490
R. Farkh, K. Laabidii & M. Ksouri Table 1. System parameters.
L K T
Step 1
Step 3
Step 5
Step 7
0.57 0.5874 1.6201
0.6 0.6256 1.5671
0.54 0.6597 1.922
0.57 0.6825 1.7265
According to this table, that the coe±cient in the following bounds 0:58 < K < 0:68, 1:56 < T < 1:92 and 0:54 < L < 0:6. Ls We consider the plant family GðsÞ ¼ Ke 1þTs where 0:58 < K < 0:68, 1:56 < T < 1:92 and 0:54 < L < 0:6. The ¯nal PI stability region that guarantee stability of GðsÞ, under parameter varying case, is included in that one obtained by using only the maximum delay L ¼ L ¼ 0:6. The entire family GE ðs; Þ is given as follows: 9 8 Gij ðs; Þ= > > > > > > > > > > l k > l K 1 ðsÞ þ ð1 l ÞK 1 ðsÞ > > > > > > > G ðs; Þ ¼ ij > > h ðsÞ > > K > > 2 > > > > S > > = < GE ðs; Þ ¼ > > K 1h ðsÞ > > > > > > ðs; Þ ¼ G ji > > l k > > K ðsÞ þ ð1 ÞK ðsÞ > > m m 2 2 > > > > > > > > > > > > 2 ½0; 1; h ¼ 1; 2; 3; 4; > > > > ; : ½l; k ¼ ½1; 2; ½1; 3; ½2; 4; ½3; 4 According to Eq. (14), we obtain: 9 8 Gij ðs; Þ= > > > > > > > > > > 1 1 > > > > ; g ; g ¼ ¼ > > 1 2 > > 1 þ 1:56s 1 þ 1:92s > > > > > > > > > > > > 1 þ 0:1 1 þ 0:1 > > > > ; g ; g ¼ ¼ > > 3 4 > > > > 1 þ 1:56s 1 þ 1:92s > > > > > > S > > > > > > = < 0:58 0:68 GE ðs; Þ ¼ ; g6 ¼ ; g ¼ > > > > > > 5 1 þ 1:56s 1 þ 1:56s > > > > > > > > > > 0:58 0:68 > > > > g ; g ; ¼ ¼ > > 7 8 > > > 1 þ 1:92s 1 þ sð1:56 þ 0:36Þ > > > > > > > > > > > 0:58 0:68 > > > g9 ¼ ; g10 ¼ ;> > > > > > > 1 þ sð1:56 þ 0:36Þ 1 þ 1:92s > > > > > > ; : 2 ½0; 1
Robust Control for Uncertain Delay System
491
We remark that g3 , g4 , g8 and g9 have an in¯nity of transfer function's sets due to their dependence in . To reduce the complexity of this problem, we set at 0, 0:2, 0:5, 0:7 and 1 as di®erent values of 2 ½0; 1. We de¯ne ghp ¼ gh ðs; p Þ where p 2 f0; 0:2; 0:5; 0:7; 1g for h ¼ 1; . . . ; 10; and p ¼ 1; . . . ; 5. So we obtain: 8 gh1 ¼ gh ðs; ¼ 0Þ ; > > > > > > > g ¼ gh ðs; ¼ 0:7Þ ; > > : h4 gh5 ¼ gh ðs; ¼ 1Þ : Figure 7 presents the stabilizing ðKp ; Ki Þ values for the entire family G by invoking the result presented in Ref. 11 which is applying for each transfer function that belongs to GE . The intersection of these stabilities regions presents an overlapped area of the boundaries which constitutes the entire feasible controller sets that stabilize the entire family GðsÞ. We choose some PI controller parameters inside the stability region given by Fig. 8 and which are presented in Table 2.
9 8 7 6
Ki
5 4 3 2 1 −2
0
2
4
6
8
10
Kp Fig. 7. The stabilizing set of ðKp ; Ki Þ for GðsÞ.
g1 g2 g3 g32 g33 g34 g35 g4 g42 g43 g44 g5 g6 g7 g82 g83 g84 g92 g93 g94 g10
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R. Farkh, K. Laabidii & M. Ksouri
5 g3 4.5 4 3.5
Ki
3 2.5 2 1.5 1 0.5 0 −1
0
1
2 Kp
3
4
5
Fig. 8. Final stability region in ðKp ; Ki Þ plan for uncertain plant GðsÞ.
Table 2. PI controllers. Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Kp
0.1
0.9
1
1
1.37
1.54
2.94
3.33
3.67
3.95
Ki
0.1
0.7
0.33
0.65
2.77
1.18
4.04
4.36
4.34
4.33
The following algorithm describes the real-time implementation of the PI control law: (1) (2) (3) (4) (5) (6)
Initialization k ¼ 1, Output acquisition yðkÞ, Error computation eðkÞ ¼ yc ðkÞ yðkÞ, Control law computation uðkÞ ¼ uðk 1Þ þ Kp eðkÞ þ ðKi Te Kp Þeðk 1Þ, Application of the control law uðkÞ to the process Exhausting of the sampling time, k ¼ k þ 1 then go to Step 2.
The systems outputs and control signals respectively presented in Figs. 918 in the case of using di®erent PI controllers as described in Table 2.
Robust Control for Uncertain Delay System
7
6
u(k),y(k) [V]
5
4
3
2
1
0
y(k) u(k) 0
1000
2000
3000 iterations
4000
5000
6000
Fig. 9. Evolution of the output and the control law (case 1).
7
6
u(k),y(k) [V]
5
4
3
2 y(k) u(k)
1
0
0
1000
2000
3000 iterations
4000
5000
Fig. 10. Evolution of the output and the control law (case 2).
6000
493
R. Farkh, K. Laabidii & M. Ksouri
7
6
y(k),u(k) [V]
5
4
3
2 u(k) y(k)
1
0 0
1000
2000
3000 iterations
4000
5000
6000
Fig. 11. Evolution of the output and the control law (case 3).
6.5 6 5.5 5 u(k),y(k) [V]
494
4.5 4 3.5 3 y(k) u(k)
2.5 2 1.5
0
1000
2000
3000 iterations
4000
5000
Fig. 12. Evolution of the output and the control law (case 4).
6000
Robust Control for Uncertain Delay System
7
6
y(k),u(k) [V]
5
4
3
2
1
0
y(k) u(k) 0
1000
2000
3000 iterations
4000
5000
6000
Fig. 13. Evolution of the output and the control law (case 5).
7
6
u(k),y(k) [V]
5
4
3
2 y(k) u(k)
1
0
0
1000
2000
3000 iterations
4000
5000
Fig. 14. Evolution of the output and the control law (case 6).
6000
495
R. Farkh, K. Laabidii & M. Ksouri
8 7 6
u(k),y(k) [V]
5 4 3 2 1 u(k) y(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
6000
Fig. 15. Evolution of the output and the control law (case 7).
8 7 6 5 u(k),y(k) [V]
496
4 3 2 1
y(k) u(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
Fig. 16. Evolution of the output and the control law (case 8).
6000
Robust Control for Uncertain Delay System
8 7 6
u(k),y(k) [V]
5 4 3 2 1
u(k) y(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
6000
Fig. 17. Evolution of the output and the control law (case 9).
8 7 6
u(k),y(k) [V]
5 4 3 2 1 y(k) u(k)
0 −1
0
1000
2000
3000 iterations
4000
5000
Fig. 18. Evolution of the output and the control law (case 10).
6000
497
498
R. Farkh, K. Laabidii & M. Ksouri
7. Conclusion In this paper, a version of the HermiteBiehler theorem and Generalized Kharitonov Theorem is adopted and proposed to ¯nd the PI stability region for the control of an uncertain ¯rst-order plant with time-delay. This stabilization problem is reduced to the equivalent problem of stabilizing a ¯nite set of 10 external plants. Also, an air stream heater has been controlled using a robust PI controller to demonstrate the e®ectiveness of proposed technique.
References 1. B. R. Barmish, C. V. Holot, F. J. Kraus and R. Tempo, Extreme points results for robust stabilization of interval plants with ¯rst-order compensators, IEEE Trans. Automat. Contr. 38 (1993) 17341735. 2. S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach (Prentice-Hall, Upper Saddle River, 1995). 3. S. P. Bhattacharyya and H. Chapellat, A generalization of Kharitonov's Theorem: Robust stability interval plants, IEEE Trans. Automat. Contr. AC-34 (1989) 306311. 4. C. Knospe and G. Editor, PID control, IEEE Contr. Syst. Mag. (2006). 5. L. Dugard and E. I. Verriest, Stability and Control for Time-Delay System (Springer, 1998). 6. M. T. Ho, A. Datta and S. P. Bhattacharyya, Design of P, PI and PID controllers for interval plants, Proc. American Control Conf. (1998). 7. Y. J. Huang and Y. J. Wang, Robust PID tuning strategy for uncertain plants based on the Kharitonov theorem, ISA Trans. 39 (2000) 419431. 8. V. L. Kharitonov, Asymptotic stability of an equilibrium position of a family of systems of linear di®erential equations, Di®er. Equat. 14 (1979) 148. 9. S.-I. Niculescu, Delay E®ects on Stability (Springer, London, 2001). 10. G. J. Silva, A. Datta and S. P. Bhattacharyya, Stabilization of time-delay systems, Proc. American Control Conf. (2000), pp. 963970. 11. G. J. Silva, A. Datta and S. P. Bhattacharyya, PI stabilization of ¯rst-order systems with time-delay, Automatica 37 (2001) 20252031. 12. G. J. Silva, A. Datta and S. P. Bhattacharyya, Stabilization of ¯rst-order systems with time-delay using the PID controller, Proc. American Control Conf. (2001), pp. 46504655. 13. G. J. Silva, A. Datta and S. P. Bhattacharyya, PID Controllers for Time-Delay Systems (Springer, London, 2005). 14. N. Tan, I. Kaya, C. Yeroglu and D. P. Atherton, Computation of stabilizing PI and PID controllers using the stability boundary locus, Energ. Convers. Manag. 47 (2006) 30453058. 15. Q. C. Zhong, Robust Control of Time-Delay System (Springer, London, 2006). 16. B. M. Patre and P. J. Deore, Robust stability and performance for interval process plants, ISA Tran. 46 (2007) 343349. 17. K. Gu, V. Kharitonov and J. Chen, Stability of Time-Delay Systems (Birkhäuser, Boston, 2003). 18. K. Gu and V. Kharitonov, Robust stability of time-delay systems, IEEE Trans. Automat. Contr. 39 (1994).
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19. V. L. Kharitonov, J. A. Torres-Muñoz and M. B. Ortiz-Moctezuma, Polytopic families of quasi-polynomials: Vertex-type stability conditions, IEEE Trans. Circuit Syst. 50 (2003). 20. M. Fu, A. W. Olbrot and M. P. Polis, Robust stability for time-delay systems: The edge theorem and graphical tests, IEEE Trans. Automat. Contr. 34 (1989) 813819. 21. V. L. Kharitonov, J. A. Torres-Muñoz and M. B. Ortiz-Moctezuma, Stability of a multidiamond type family of quasi-polynomials, Proc. 47th IEEE Conf. on Decision and Control. 22. S. Mondie, J. Santos and V. L. Kharitonov, Robust stability of quasi-polynomials and the ¯nite inclusions theorem, IEEE Trans. Automat. Contr. 50 (2005). 23. V. A. Oliveira, L. V. Cossi, C. M. Teixeira and A. M. F. Silva, Synthesis of PID controllers for a class of time-delay systems, Automatica, doi:10.1016/j.automatica.2009.03.018. 24. D. J. Wang, Synthesis of PID controllers for high-order plants with time-delay, J. Process Control (2009). 25. J. Fang, D. Zheng and Z. Ren, Computation of stabilizing PI and PID controllers by using Kronecker summation method, Energ. Conver. Manag. 50 (2009) 18211827. 26. M. S. Mahmoud, Stabilization of time-delay systems by PID controllers, Int. Multi-Conf. Systems, Signals and Devices, SSD'09. 27. G. J. Silva, A. Datta and S. P. Bhattacharyya, Robust control design using PID controller, IEEE Conf. Decision and Control (2002).
