A low-cost benchmark on hybrid control of thermal heat and power plants.
Key words: Hybrid control, Control benchmarks
Hybrid systems have evolved into a hot topic among control theorists during the last decade. It is not difficult to find out why. Many industrial plants and processes have neither purely continuous nor purely discrete (or in a special case logical) dynamics but they include both continuous valued and discrete valued variables and elements (on/off switches or valves, speed selectors). Many plants exhibit different dynamic behaviours in different modes of operation. Such plants are modelled as a combination of several continuous models and discrete valued variables that determine which of these models is valid under the current operation conditions. Thus the theory of hybrid systems that includes both continuous and discrete valued dynamics and variables in one general framework is of great practical importance. Many important results concerning hybrid systems have been achieved (Christofides & El-Farra, 2005; Savkin & Evans, 2002). However, this field still mostly remains the topic of highly theoretical papers. The testing of hybrid control approaches is usually done merely in simulation. Occasionally some papers sketch simple model plants. These plants are used to illustrate the ideas described in the paper but they have never been built and used for real testing (e.g., Lennartson et al, 1996). Still other papers include hybrid control or modelling experiments with laboratory models. However, although these models exhibit certain hybrid phenomena, they were originally designed for continuous control experiments (e.g. Kowalewski et al., 1999).
Just recently there has been some shift to more realistic benchmarks for testing of hybrid control algorithms. In particular, the big European project HYCON focused on hybrid control (www.ist-hycon.org) lays considerable emphasis on control performance benchmarking and using physical models for this purpose. However, even in the framework of this project just one physical model has been built: a mechanical model with backlash (see Besselmann et al., 2007). Other benchmarks are just simulation models or unrealized proposals.
This paper describes another benchmark plant. Unlike the one proposed in (Besselmann et al., 2007) it is not a mechanical system but it is designed in such a way that it exhibits hybrid phenomena that are typical of thermal heat and power processes. The benchmark plant has been developed and built within the framework of a joint project of the CTU Prague and Technical University of Liberec supported by the Czech Science Foundation. The plant is relatively simple and inexpensive to build. This paper outlines the structure of the plant, gives its mathematical model and control experiments that can be performed with the plant and associated continuous-discrete interactions that represent a considerable challenge for control design. This plant can also be used for educational purposes. A detailed treatment of possible educational applications of this plant was given in (Hlava et al., 2006).
2. BENCHMARK LABORATORY SCALE PLANT
The structure of plant is shown in Fig. 1. As in most process control applications, the measured and controlled variables are water levels, temperatures and flows. Although this plant does not represent any particular process exactly, both the plant itself and its instrumentation are well representative of plants commonly used in the process industries. The basic components of the plant are three water tanks. Water from the reservoir can be drawn by pumps 1 and 3 to the respective tanks. The delivery rate is measured with a turbine flowmeter. The flow from pump 3 is fed directly to tank 3. The flow from pump 1 goes through a storage water heater and it is further controlled by an on/off solenoid valve S1. The temperature at the heater output is measured with a Pt1000 sensor TT4. The power consumption of the heater can be changed continuously. Another heater is mounted at the bottom of tank 2. Depending on the selected control scenario, the inflows to tanks 1 and 3 can be used as manipulated variables or as disturbances. Tanks are thermally insulated to make the heat losses negligible. Tanks 2 and 3 have a special shape that introduces a change of the dynamics at a certain level. Their behaviour is described by hybrid switched models.
3. MATHEMATICAL MODEL
Model is derived by putting together mass and energy balance equations of each single tank and using Bernoulli equation for the flow from tank 1 to tank 2. Model is given by
[[??].sub.1] (t) = (1/[A.sub.1]) ([q.sub.1](t)-0.1 [k.sub.v][[sigma].sub.1](t)[square root of g[h.sub.1](t)]) (1)
[[??].sub.1] (t) = [q.sub.1](t)([v.sub.4](t)-[v.sub.1](t))/[A.sub.1][h.sub.1](t) (2)
[FIGURE 1 OMITTED]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where P is power output of heater [H.sub.2], [A.sub.i] = [pi][r.sup.2.sub.i], [DELTA]r = [r.sub.31] - [r.sub.32] and the meaning of other variables (levels, temperatures, flows) is evident from Fig. 1. Discrete state variables are denoted [m.sub.1] and [m.sub.2], discrete valued input [[sigma].sub.1] assumes values 0,1,2 (no valve open, S3 open, S3 and S4 open), [[sigma].sub.2] assumes values 0, 1, 2, 3 (pump 4 delivery rate is 0, 1/3 max., 2/3 max., maximum), [q.sub.0] is the delivery rate of pump 4 running at 1/3 of maximum.
Vector field defined by (1) to (5) involves both controlled and autonomous switching. Controlled switching due to changes of discrete valued inputs [[sigma].sub.1], [[sigma].sub.2] results in vector field discontinuity in (1), (3), (4) and (5). Autonomous switching due to changes of [h.sub.2] and [h.sub.3] results in vector field discontinuity in (3). The dynamic behaviours of (4) and (5) are also switched, but the vector field remains continuous. Evidently this model is of the same form as general state equations of a hybrid system
[??](t) = f (x(t), m(t), u(t)) (8)
y(t) = g (x(t), m(t), u(t))
m([t.sup.+]) = [phi] (x(t), m(t), u(t), [sigma](t)) (9)
o([t.sup.+]) = [phi] (x(t), m(t), u(t), [sigma](t))
x([t.sup.+]) = [psi] (x(t), m(t), u(t), [sigma](t)) (10)
The only feature that is not present are state jumps described by (10). However they are typical of mechanical systems only.
4. CONTROL EXPERIMENTS
The plant allows us to define many control scenarios for the evaluation of hybrid control approaches. Some are outlined below, many other scenarios are possible. First, uncomplicated control tasks can be defined, e.g. water level control in tank 2 or 3 or temperature control in tank 2. Even then the controlled system features switched dynamics For instance, if the control objective is to control [h.sub.2] using [q.sub.1] as a manipulated variable (valve S2 open, S1, S3, S4 closed) and [q.sub.2] is a disturbance (in this scenario both [q.sub.1] and [q.sub.2] can be changed in several steps) the controlled system is a integrator hybrid system. Alternatively, the objective can be water level control in tank 3. The dynamics of (5) are nonlinear for [h.sub.3] [less than or equal to] [l.sub.2]. The nonlinear range of [h.sub.3] can be divided into several sub-ranges and the whole system including switching and non-linearity can be approximated with a hybrid piecewise affine (PWA) system.
More complicated scenarios use two tanks. One scenario can be formulated as follows. Tank 1 is a buffer that receives water from an upstream process. Water flow rate and temperature are disturbances. The control objective is to deliver the water to a downstream process at a desired temperature. The flow demand of the downstream process is another disturbance. Valves S1, S2 and S3 are discrete valued manipulated variables and the power output of heater [H.sub.2] is a continuous manipulated variable. The main control objective necessarily includes several auxiliary objectives. Tank levels must be kept within specified limits, and overflow as well as emptying must be avoided. The controlled system is hybrid and nonlinear. Unlike the previous simpler scenarios it includes nontrivial interactions of continuous and discrete controls. For example, if S3 is already open and S4 must also be opened to avoid tank 1 overflow, temperature [v.sub.2] may also be significantly affected, particularly if [v.sub.1] and [v.sub.2] are much different. The plant in this configuration can be used to test approaches to hybrid system control. It is possible to follow the traditional approach and to design the water level control logic and continuous heater control independently. Alternatively, a hybrid approach can be followed and both the discrete and the continuous manipulated variables can be controlled with a single controller designed using hybrid control methods. The performance achieved with both approaches can be compared. Even more complicated interactions can occur in scenarios that use all three tanks.
The laboratory-plant outlined in this paper exhibits hybrid phenomena typical of process control applications, in particular in the field of thermal heat and power processes. Thus it can be used as a suitable benchmark for evaluation of the performance of various approaches to the control of hybrid systems. The variety of possible control scenarios outlined in the previous sections demonstrates the considerable flexibility of the plant.
This research has been supported by the Czech Science Foundation within project 101/07/1667.
Besselmann, T.; Rostalski, P.; Barie, M.; Lagerberg, A. & Morari, M. (2007). A Benchmark on Hybrid Control of a Mechanical System with Backlash, Benchmark proposal for HYCON, Network of Excellence in Hybrid Control, Automatic Control Laboratory, ETH, Zurich, Switzerland
Christofides, P.D.,& El-Farra, N. (2005). Control of Nonlinear and Hybrid Process Systems Springer, ISBN: 978-3-540-28456-7
Hlava, J., Sulc, B. & Tamas, J. (2006). A Laboratory Scale Plant with Hybrid Dynamics and Remote Access via Internet for Control Engineering Education, Proceedings of the 16th World Congress of the International Federation of Automatic Control, Elsevier, ISBN: 978-0-08-045108-4
Kowalewski S. et al. (1999), A Case Study in Tool-Aided Analysis of Discretely Controlled Continuous Systems: The Two Tanks Problem, In: Lecture Notes in Computer Science, Vol. 1567, 1999, pp. 163-185.
Lennartson, B.; Titus, M.; Egardt B. & Pettersson, S. (1996) Hybrid Systems in Process Control, IEEE Control Systems Magazine, Vol. 16, No. 5, pp. 45-56.
Savkin A.V. & Evans R.J. (2002). Hybrid Dynamical Systems--Controller and Sensor Switching problems. Birkhauser, ISBN 0-8176-4224-2, Boston
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|Publication:||Annals of DAAAM & Proceedings|
|Article Type:||Technical report|
|Date:||Jan 1, 2007|
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