Robust Control of a Two Tank System Using Algebraic Approach Marek Dlapa*, Roman Prokop, and Monika Bakosova Tomas Bata University in Zlin Nad Stranemi 4511, 760 05 Zlin, Czech Republic {dlapa,prokop}@fai.utb.cz,
[email protected] Abstract. The paper deals with design of a robust controller via algebraic µ-synthesis for a two tank system, which is a well known benchmark problem. The controller is obtained by decoupling two-input two-output system into two identical SISO (Single-Input Single-Output) plants. The task of robust controller design is then performed by finding a suitable pole placement for the SISO systems. The robustness is measured by the structured singular value denoted µ. The final controller is verified through simulation for plants perturbed by worst case perturbations.
1 Introduction Algebraic methods ([7], [11]) are well known and easy to use for SISO (single-input single-output systems) systems described by continuous or discrete transfer functions. However, if applied to MIMO (multi-input multi-output) systems computational difficulties are increasing. In this paper, the problem of MIMO system design is treated via decoupling the MIMO system into two identical SISO plants, which are approximated by transfer functions with simple structure. This guarantees decoupled result control, and simplifies derivation of pole placement formulae. In order to measure the robust stability and performance, the structured singular value denoted μ is used ([5], [6]). The algebraic μ-synthesis [2] overcomes some difficulties connected with D-K iteration, namely the fact that it does not guarantee convergence to a global or even local minimum, which leads to non-optimality of the resulting controller [10]. Moreover, controllers obtained via algebraic approach can have simpler structure as there is no need of absorbing the scaling matrices into generalized plant, and hence no need of further simplification causing deterioration of frequency properties of the controller. In this paper, the algebraic μ-synthesis is applied to the control of a two tank system [1], which is a well known benchmark for the robust control design. The following notation is used: || ⋅ ||∞ denotes H∞ norm, Rps is the ring of Hurwitz-stable and proper rational functions, and In is the unit matrix of dimension n.
2 Model Description Detailed description of the two tank system is in [1] and related papers (e.g. [8] and [9]). Here only points important for the proposed method are presented. *
Corresponding author.
R. Moreno-Díaz et al. (Eds.): EUROCAST 2009, LNCS 5717, pp. 603–609, 2009. © Springer-Verlag Berlin Heidelberg 2009
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The system consists of two water tanks in cascade depicted in Fig. 1. The upper tank (tank 1) is fed by hot and cold water via computer controllable valves. The lower tank (tank 2) is fed by water from an exit at the bottom of tank 1. A constant level is maintained in tank 2 by means of overflow. A cold water bias stream also feeds tank 2 and enables the tanks to have different steady-state temperatures.
Fig. 1. Schematic diagram of the two tank system
From the brief description follows that the two tank system is a MIMO plant with two measured signals: t1, t2 and two inputs fh and fc. The quantities t1, t2 represent temperatures of tank 1 and 2 respectively. The input signals are commands to hot flow (fhc) and cold flow (fcc) actuators, which are transformed to hot water flow fh and cold water flow fc. Third measured signal is water level in tank 1 (h1), which is not controlled. This quantity is, however, important for assessment of controller performance. Due to linearization quantities h1 and t1 should be in a prescribed range: 0.25 ≤ h1≤ 0.75
(1)
0.25 ≤ t1≤ 0.75
(2)
Nominal model from the inputs fh, fc to the outputs t1, t2 can be written in two dimensional transfer matrix ⎡ P ( s) P12 ( s ) ⎤ Pnom ( s ) ≡ ⎢ 11 ⎥≡ ⎣ P21 ( s) P22 ( s )⎦ ⎤ ⎡ 0.0036s 2 + 0.0001s + 7.8157 ⋅ 10 - 7 0.0004s + 4.6643 ⋅ 10 - 6 ⎢ 3 2 -6 3 2 -6 ⎥ s s s s s s 0.0491 0.0007 3.0684 10 0.0491 0.0007 3.0684 10 + + + ⋅ + + + ⋅ ≡⎢ ⎥ 2 -6 0.1562 ⋅ 10 - 4 s + 0.0187 ⋅ 10 - 4 ⎥ ⎢ - 0.0109s - 0.0004s - 2.3447 ⋅ 10 ⎢⎣ s 3 + 0.0491s 2 + 0.0007s + 3.0684 ⋅ 10 - 6 s 3 + 0.0491s 2 + 0.0007s + 3.0684 ⋅ 10 - 6 ⎥⎦
(3)
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The differences from the nominal model are treated by multiplicative perturbations at the outputs of measured quantities (Fig. 2). Perturbation weights are transfer functions of the form: 0.5s 0.25s + 1
(4)
20hˆ1 s 0 .2 s + 1
(5)
100s s + 21
(6)
W h1 = 0.01 +
W t 1 = 0 .1 +
Wt 2 = 0.1 +
here hˆ1 = 0.75 is steady state value of h1. For details on nominal models of tank 1 and 2 see [1]. Sensor noise is modelled by adding weighted unknown input to h1, t1, and t2. The appropriate weights are Wt1noise = 0.03
(7)
Wt2noise = 0.03
(8)
Fig. 2. Schematic representation of the perturbed, linear, two tank model
3 Algebraic μ-Synthesis For the purposes of algebraic design the MIMO system is decoupled into two identical SISO plants. For decoupling of the nominal plant Pnom it is satisfactory to have the controller in the form
K ( s ) = K ( s)I 2 det[ Pnom ( s )]
1 [ Pnom ( s)] −1 P11 ( s)
(9)
The choice of decoupling matrix prevents controller from cancelling any poles or zeroes in the right half-plane so that the internal stability of the nominal feedback
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loop is held. The MIMO problem is now transformed into finding a controller K(s), which is tuned via setting the poles of the nominal feedback loop with the plant
Pdec (s) =
1 1 det[Pnom (s )][Pnom ( s)] −1 Pnom ( s) = det[Pnom ( s)]I 2 P11 ( s) P11 (s )
(10)
Let
Pdec ≡
1 det[Pnom ( s)] P11 ( s )
(11)
Then the transfer function Pdec can be approximated by 2nd order system ∗ Pdec ( s) =
bdec ( s ) 0.0027 s 2 - 0.0017 s + 0.0001 = a dec ( s ) s 2 + 0.0164 s + 0.0001
(12)
and the controller K = N/M is obtained by solving the Diophantine equation (13)
AdecM + BdecN = 1
with Adec, Bdec, M, N ∈ Rps. It follows from the algebraic theory that the asymptotic tracking of the reference signal is achieved if and only if AdecM is divisible by Fr and disturbance is suppressed if AdecM is divisible by Fd. Here Fr and Fd are Laplace transforms of the reference and disturbance respectively. By the analysis of polynomial degrees of adec and bdec transfer functions Adec, Bdec, M and N are chosen so that the number of closed loop poles is minimal and the asymptotic tracking for step reference signal is achieved: Adec =
a dec 2
∏ (s + α )
, Bdec =
sm
∏ (s + α )
(14)
i =1
i =1
4
∏ (s + α ) i
i
M =
bdec 2
, N=
i
n 4
∏ (s + α )
(15)
i
i =3
i =3
Degrees of polynomials m and n are: ∂m = 1, ∂n = 2
(16)
Thus the characteristic polynomial of the nominal closed loop has 4 pairs of poles –αi, which represent the tuning parameters. The resulting controller is
K ( s) =
n sm
(17)
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The open-loop interconnection is in Fig. 3 with performance and reference weights
Wt1 perf =
130 s + 1 130s + 0.5 , Wt 2 perf = 400 s + 1 800s + 1
(18)
Wt1cmd = 1 , W t
(19)
diffcmd
=1
0 ⎤ ⎡ w11 ⎤ ⎡t1 ⎤ ⎡Wt1noise 0 ⎤ ⎡ w21 ⎤ 0 ⎤ ⎡Wt1cmd ⎢ 0 W ⎥⎢ ⎥ − ⎢ ⎥ − ⎢ ⎥⎢ ⎥ ⎥ 1 ⎦ ⎢⎣ t diffcmd ⎥ ⎦ ⎣w12 ⎦ ⎣t2 ⎦ ⎣ 0 Wt 2 noise ⎦ ⎣w22 ⎦
⎡ t1err ⎤ ⎡ 1 ⎢t ⎥ = ⎢ ⎣ 2err ⎦ ⎣ 1
(20)
Controller inputs are tracking errors of measured temperatures t1err, t2err. The relaxed performance weight is justified by additional postulate of decoupled result control for the nominal loop, which is not common in standard design, and which makes the task of achieving the robust performance and stability more difficult. This modification of the interconnection does not degrade the uncertainty model. The resulting performance can be observed by simulations for nominal and perturbed plants.
reference weights
v
e1
error weights ⎡ t1err ⎤ ⎢t ⎥ ⎣ 2 err ⎦
Δ
z
pert. model
+
+
w1
noise weights
+ e2
⎡ f hc ⎤ ⎢ ⎥ ⎣ f cc ⎦
actuator model
w2
actuator weights G
Fig. 3. Interconnection structure for algebraic approach
The controller design is then reduced to minimization of the peak of the μ-function. The cost function is defined as sup μ Δ~ [Fl (G, K )] ω∈R
(21)
~ Here Δ denotes augmented perturbation including performance weights in feedback loop. In order to overcome the multimodality of the cost function an evolutionary algorithm Differential Migration (e.g. [3]) was used for searching optimal values of αi. Rough results obtained from DM were tuned up by the Nelder-Mead simplex method. Poles were constrained to the interval of -300 to 0.
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Simulations for worst case perturbations are in Fig. 4 and 5. It can be observed that no steady state error is present, and response to ramp reference signal is monotonous, which is not true for the D-K iteration. Polynomial approach, step response for perturbed plant
Polynomial approach, step response for perturbed plant
0.9
0.75
fhc fcc 0.8
0.7
Actuators
Measurements
0.7
h1 t1 t2
0.65
0.6
0.6
0.5
0.55
0.4 0.5
0.45
0.3
0
200
400
600
800 1000 1200 Time: seconds Time (seconds)
1400
1600
1800
0.2
2000
0
200
400
600
800 1000 1200 Time: seconds Time (seconds)
1400
1600
1800
2000
Fig. 4. Response to reference signal which ramps (from 80 to 100 seconds) t1 from 0.75 to 0.57, and t2 from 0.67 to 0.47 for algebraic approach and perturbed plant Design k0, step response
Design k3, step response (perturbed plant) 0.9
0.8
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0.8
0.7
0.7
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Actuators
Measurements
fhc fcc
h1 t1 t2
0.6
0.5
0.55
0.4
0.5
0.3
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0.2 0
1000
2000
3000
4000 5000 6000 Time: seconds Time (seconds)
7000
8000
9000 10000
0
1000
2000
3000
4000 5000 6000 Time: seconds Time (seconds)
7000
8000
9000 10000
Fig. 5. Response to reference signal which ramps (from 80 to 100 seconds) t1 from 0.75 to 0.57, and t2 from 0.67 to 0.47 for D-K iteration and perturbed plant
4 Conclusion The paper has presented another application of the algebraic approach to a MIMO system. The plant has been decoupled into two identical SISO systems and the controller has been designed via optimization of nominal plant poles. The performance and robustness have been measured by supremum of the μ-function in frequency domain. Besides its simpler structure, the resulting controller satisfies robust performance condition and guarantees the robust stability. Simulation proved that the algebraic approach has monotonous response and fast set point tracking for ramp reference signal. Moreover, the asymptotic tracking is achieved, which is not held for
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the reference method. The better performance of the controller is due to the fact that the algebraic method implements decoupled control for the nominal closed loop. This scheme cannot be used in the scope of the D-K iteration. The D-K iteration yields higher robustness as it fully utilizes the MIMO structure of the controller, and makes a trade-off between robust stability and performance. However, the higher stability is achieved at the expense of worse performance. Although it is apparent that generally the presented approach cannot substitute the standard method, in this case, the algebraic design proves better performance than the standard procedure, and confirms the results obtained in [4].
Acknowledgment This work was supported by the project MSM7088352102.
References 1. Balas, G.J., Doyle, J.C., Glover, K., Packard, A., Smith, R.: μ- Analysis and Synthesis Toolbox for Use with MATLAB. The MathWorks, Inc. (1998) 2. Dlapa, M., Prokop, R.: μ-Synthesis: Simple Controllers for Time Delay Systems. In: Proceedings of 11th Mediterranean Conference on Control and Automation, Rhodes, Greece (2003) 3. Dlapa, M., Prokop, R.: Differential Migration: a new algorithm for global optimization. In: Proceedings of the 6th Asian Control Conference, pp. 618–626 (2006); ISBN 979-15017-0 4. Dlapa, M., Prokop, R.: Control of the HIMAT Aircraft via Algebraic μ-Synthesis. In: Proceedings of the 7th Portuguese Conference on Automatic Control (2006); ISBN 978972-97025-2-5 5. Doyle, J.C.: Analysis of Feedback Systems with Structured Uncertainties. Proceedings of IEEE, Part-D 129, 242–250 (1982) 6. Doyle, J.C.: Structure Uncertainty in Control System Design. In: Proceedings of 24th IEEE Conference on Decision and Control, pp. 260–265 (1985) 7. Kučera, V.: Discrete Linear Control: The Polynomial Equation Approach. Wiley, New York (1972) 8. Smith, R.S., Doyle, J., Morari, M., Skjellum, A.: A case study using μ: Laboratory process control problem. In: Proc. Int. Fed. Auto. Control, vol. 8, pp. 403–415 (1987) 9. Smith, R.S., Doyle, J.: The two tank experiment: A benchmark control problem. In: Proc. Amer. Control Conf., vol. 3, pp. 403–415 (1988) 10. Stein, G., Doyle, J.: Beyond Singular Values and Loopshapes. AIAA Journal of Guidance and Control 14(1), 5–16 (1991) 11. Vidyasagar, M.: Control Systems Synthesis: A Factorization Approach. MIT Press, Cambridge (1985)
