Ege Üniversitesi Elektrik Elektronik Mühendisliği Bölümü Kontrol Sistemleri II Dersi Grup Adı: Sıvı Seviye Kontrol Deneyi.../../2015 KP Pompa akış sabiti 3.3 cm3/s/v DO1 Çıkış-1 in ağız çapı 0.635 cm DO2 Çıkış-2 nin ağız çapı 0.47625 cm A t1 Tank-1 in kesit alanı 15.5179 cm 2 A t2 Tank-2 nin kesit alanı 15.5179 cm 2 g Yer çekimi sabiti 981 cm/s 2 L 10 Seviye-1 in denge noktası 15 cm L 20 Seviye-2 nin denge noktası 15 cm V MAX_DAC Maksimum verilebilecek voltaj 10 volt A o1 A o2 F i1 F i2 F o1 F o2 Tank-1 çıkış deliğinin alanı Tank-2 çıkış deliğinin alanı Tank-1 e suyun akış hızı Tank-2 e suyun akış hızı Tank-1 den suyun çıkış hızı Tank-1 den suyun çıkış hızı GİRİŞ Bu deneyde şekilde görülen sıvı seviye tank sisteminin kontrolü gerçekleştirilmeye çalışılacaktır. İlk etapta tank 1 in sıvı seviye kontrolü PID denetleyici gerçekleştirilecek, daha sonra da tank 2 nin sıvı seviye kontrolü PI denetleyici ile gerçekleştirilecektir. Bu deneyde yapılacak işlemler aşağıda sıralanmıştır; Matematiksel hesaplama ile elde edilen doğrusallaştırılmış sistem modellerini kullanarak, ön çalışmadaki istenen özellikleri sağlayacak PID parametrelerini Tank1 Seviye kontrol ve Tank2 Seviye Kontrol için ayrı ayrı bulmak ve not etmek. Elde edilen PID parametrelerini kullanarak, doğrusal model ile doğrusal olmayan modelin basamak cevabını karşılaştırmak. Doğrusal olmayan model ile gerçek modelin basamak cevabını karşılaştırmak.
MODELLING where A t1 is the area of Tank 1. F i1 and F o1 are the inflow rate and outflow rate, respectively. The volumetric inflow rate to tank 1 is assumed to be directly proportional to the applied pump voltage, such that: Applying Bernoulli's equation for small orifices, the outflow velocity from tank 1, v o1, can be expressed by the following relationship: A o1 is the cross-section area of tank 1 outlet hole: The nonlinear equation of motion (EOM) of tank 1 should be linearized around a quiescent point of operation. By definition, static equilibrium at a nominal operating point (V p0,l 10) is characterized by the Tank 1 level being at a constant position L 10 due to a constant water flow generated by constant pump voltage V p0. In the case of the water level in tank 1, the operating range corresponds to small departure heights, L 11, and small departure voltages, V p1, from the desired equilibrium point (V p0,l 10). Therefore, L 1 and V p can be expressed as the sum of two quantities, as shown below: The obtained linearized EOM should be a function of the system's small deviations about its equilibrium point (V p0,l 10). For a function, f, of two variables, L 1 and V p, a first-order approximation for small variations at a point (L 1,V p) =(L 10,V p0) is given by the following Taylor's series approximation:
At equilibrium point, F i1-f o1 equals to zero. The desired open-loop transfer function for the Coupled-Tank's tank 1 system is the following: where K dc1 is the open-loop transfer function DC gain, and τ 1 is the time constant. As a remark, it is obvious that linearized models, such as the Coupled-Tank tank 1's voltage-to-level transfer function, are only approximate models. Therefore, they should be treated as such and used with appropriate caution, that is to say within the valid operating range and/or conditions. However for the scope of this lab, the Coupled-Tank tank 1's voltage-to-level transfer function is assumed valid over the pump voltage and tank 1 water level entire operating range, Vp_peak and L1_max, respectively. In deriving the tank #2 EOM the mass balance principle can be applied to the water level in tank 2 as follows
At equilibrium point, F i2-f o2 equals to zero.
PRE-LAB EXPERIMENTS TANK 1 LEVEL CONTROL Here a control is designed to regulate the water level (or height) of tank #1 using the pump voltage. The control is based on a PID. The control scheme is based on PID, as illustrated in Figure, below. Given a ±1 cm square wave level setpoint (about the operating point), the level in tank 1 should satisfy the following design performance requirements: 1. Tank 1 operating level at 15 cm: L 10 = 15 cm. 2. Percent overshoot should be less than or equal to 10%: PO 1 10:0 %. 3. Settling time less than 30 seconds: t s_1 30 s. 4. Steady-state error less than 1%: e ss_1 < 0.01 cm. 5. Input voltage should be less than 45 V: Vp<45 V. Simulation Results L1 Vp For PID gains of: Kp= Ki= Kd= N=
TANK 2 LEVEL CONTROL The pump feeds tank 1 and tank 1 feeds tank 2. The designed closed-loop system is to control the water level in tank 2 (i.e. the bottom tank) from the water flow coming out of tank 1, located above it. The control scheme is based on PI, as illustrated in Figure, below. In response to a desired ±1 cm square wave level setpoint from tank 2 equilibrium level position, the water height behaviour should satisfy the following design performance requirements: 1. Tank 2 operating level at 15 cm: L 20 = 15 cm. 2. Tank 1 operating level should be less than 25: L 1<25 cm 3. Percent overshoot should be less than or equal to 10%: PO 2 10:0 %. 4. Settling time less than 140 seconds: t s_2 140:0 s. 5. Steady-state error less than 1%: e ss_2 < 0.01 cm. 6. Input voltage should be less than 45 V: Vp<45 V. Simulation Results L1 Vp L2 For PI gains of: Kp= Ki=
LAB EXPERIMENTS TANK 1 LEVEL CONTROL 1. Compare Linear and Non-linear model responses:
2. Compare Non-linear and Real model responses:
TANK 2 LEVEL CONTROL 1. Compare Linear and Non-linear model responses:
2. Compare Non-linear and Real model responses:
SONUÇ 1. Yaptığımız uygulamalarda PI veya PID denetleyicilerinin Kp,Ki ve Kd değerlerini Ziegler-Nichols metoduyla bulup kullanabilir miydik? Nedeniyle beraber açıklayınız. 2. Tank 2 Level Control örneğinde neden PID değil de PI kullandık? 3. Denetleyici ve sistemi değiştirmeden, referans olarak 15 değil de 20 verseydik, nasıl bir sonuç elde ederdik?