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Controller Design for a Two Tank System Using Structured Singular Value and Direct Search Methods
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Controller Design for a Two Tank System Using Structured Singular Value and Direct Search Methods M. Dlapa* *Tomas Bata University in Zlin, Faculty of Applied Informatics, nám. T.G.Masaryka 5555, 760 01 Zlín Czech Republic (e-mail: [email protected]; Fax: +420 57 603 5279). Abstract: The aim of this paper is to show an algebraic approach to controller design using the structured singular value (denoted μ) as a robust stability and performance indicator. The algebraic μ-synthesis is applied to a two-tank system, which is a well known benchmark problem. The design is performed using decoupling of the nominal plant into two identical SISO (single-input single-output) systems. Global optimization methods are used due to multimodality of the cost function. The overall performance is verified by simulations for the nominal and perturbed system and compared with the D-K iteration. Keywords: Robust control; evolutionary computation; algebraic approaches; structured singular value; process control. 1. INTRODUCTION The algebraic theory (e.g. Kučera, 1972, 1993; Vidyasagar, 1985) is well known and its importance is growing due to the simplicity of controller derivation and the fact that some crucial properties of the resulting feedback loop can be easily influenced by the choice of the controller structure, which is not hard to do within the scope of this approach. The structured singular value denoted µ (see Doyle, 1982, 1985) provides a measure of robust stability and performance that can take into account many aspects of controller design including sensor noise, dynamic perturbations as well as parametric uncertainties if the linear fractional transformation (LFT) can be used for its treatment. Standard tools for µ-synthesis provide a methodology for controller design with decoupling using model matching (see Prempain and Bergeron, 1998). However, the need of defining the internal model brings another step into the process of obtaining the controller. The algebraic approach provides the methodology for synthesis of very simple controllers (PI, PID) without an additional model matching design, which simplifies controller derivation. Due to the multimodality of the cost function, an algorithm for global optimization is used for tuning nominal closed-loop pole placement, where the peak of the µ-function in frequency domain gives the measure of controller stability and performance. Knowing this, behavior of the resulting feedback system can be simulated for the worst-case perturbation causing the highest value of µ. Algebraic methods are easy to apply to SISO (single-input single-output) systems described by continuous or discrete transfer functions. Still, if used for MIMO (multi-input multioutput) systems, computational difficulties are severe. In this paper, the problem of the MIMO system design is solved via decoupling a MIMO system into two identical SISO plants, 978-3-902661-93-7/11/$20.00 © 2011 IFAC
which are then approximated by transfer functions with a simple structure. This guarantees decoupled control and simplifies derivation of the pole placement formulae. The algebraic μ-synthesis (Dlapa et al., 2009; Dlapa and Prokop, 2010) overcomes some difficulties connected with the D-K iteration, namely the fact that it does not guarantee convergence to a global or even local minimum (Stein and Doyle, 1991). Controllers obtained via the algebraic approach can have simpler structure due to the fact that there is no need of scaling matrices absorbance into the generalized plant, and hence no need of further simplification causing deterioration of the frequency properties of the resulting controller. Moreover, the controller structure can be chosen in advance, which is not possible in the scope of currently used methods. In this paper, the algebraic μ-synthesis is applied to the control of a two-tank system - classical example of robust control design. The D-K iteration is used as a reference method and the results are compared through simulations for the nominal and perturbed plant.
The following notation is used: || ||f denotes Hf norm, R and Cnum are real numbers and complex matrices, respectively, In is the unit matrix of dimension n and RPS denotes the ring of Hurwitz-stable and proper rational functions. 2. PRELIMINARIES
Define ' as a set of block diagonal matrices
Δ { {diag[G 1 I r1 , , G S I rS , '1 , , ' F ] : G i C, ' j C
where
m j um j
} (1)
S is the number of repeated scalar blocks, F is the number of full blocks, r1,}, rS and m1,}, mF are positive integers defining dimensions of scalar and full blocks.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
For consistency among all the dimensions, the following condition must be held
¦r ¦m S
i 1
F
i
j 1
j
n
(2)
Definition 1: For M C
nun
is μ'(M) defined as
1 P Δ (M ) { min{V (' ) : ' Δ, det(I M' )
0}
SISO plants. The nominal model is defined in terms of transfer functions: ª P11 ( s ) P1l ( s ) º Pnom ( s ) { «« »» ¬« Pl1 ( s ) Pll ( s ) ¼»
(7)
For decoupling the nominal plant Pnom (Pnom invertible) it is satisfactory to have the controller in the form
(3)
(8)
K ( s)
Consider a complex matrix M partitioned as
where Pxy is an element of adj[Pnom(s)] = det[Pnom(s)][Pnom(s)]–1 with the highest degree of numerator {adj[Pnom(s)] denotes adjugate matrix of Pnom}. The choice of the decoupling matrix prevents the controller from cancelling any poles or zeros from the right half-plane so that internal stability of the nominal feedback loop is held. The MIMO problem is reduced to finding a controller K(s), which is tuned via setting the poles of the nominal feedback loop with decoupled plant
M
ª M11 M12 º « » ¬M 21 M 22 ¼
(4)
and suppose there is a defined block structure '2 which is compatible in size with M22 (for any '2 '2, M22'2 is square). For '2 '2, consider the following loop equations e = M11d + M12w z = M21d + M22w (5) w = ' 2z If the inverse to I – M22'2 exists, then e and d must satisfy e = FL(M, '2)d, where
FL(M, '2) = M11 + M12'2(I – M22'2) M21 (6) is a linear fractional transformation on M by '2, and in a feedback diagram appears as the loop in Fig. 1. –1
The subscript L on FL pertains to the lower loop of M and is closed by '2. An analogous formula describes FU(M, '1), which is the resulting matrix obtained by closing the upper loop of M with a matrix '1 '1. e z
Define Pdec {
1 det[Pnom ( s )] Pxy ( s )
Pdec (s )
* Pdec (s)
Fig. 1. LFT interconnection. 3. ALGEBRAIC μ-SYNTHESIS The algebraic μ-synthesis can be applied to any control problem that can be transformed to the loop in Fig. 2, where G denotes the generalized plant, K is the controller, 'del is the perturbation matrix, r is the reference and e is the output.
errors
(10)
b( s ) a(s)
(11)
b0 b1s bn s n (D 1 s)(D 2 s) (D n s) s n a0 a1 s an1 s n1 (D 1 s)(D 2 s) (D n s)
B , A, B RPS A
(12)
The controller K = NK/DK is obtained by solving the Diophantine equation
ADK + BNK = 1 (13) with A, B, DK, NK Rps. Equation (13) is often called the Bezout identity. All feedback controllers NK/DK are given by K
Δdel
G
(9)
* Transfer function Pdec can be approximated by a system Pdec with lower order than Pdec
w '
e
1 det[Pnom ( s )][Pnom ( s )]1 Pnom ( s ) Pxy ( s ) 1 det[Pnom ( s )]I l Pxy ( s )
Pdec ( s )
which can be rewritten in terms of its coefficients and transformed to the elements of RPS
d
M
K ( s)I l det[Pnom ( s)]
1 [Pnom ( s )]1 Pxy ( s )
If no such ' ' exists making I – M' singular, then μ'(M) = 0.
NK DK
N K 0 AT DK 0 BT
,
N K 0 , DK 0 R PS
(14)
where N K 0 , DK 0 R PS are particular solutions of (13) and T is
r
an arbitrary element of RPS.
controls K
Fig. 2. Closed loop interconnection. For the purposes of the algebraic μ-synthesis, the MIMO system with l inputs and l outputs has to be decoupled into l identical
The controller K satisfying equation (13) guarantees the BIBO (bounded input bounded output) stability of the feedback loop in Fig. 3. This is a crucial point for the theorems regarding the structured singular value. If the BIBO stability is held, then the nominal model is internally stable and theorems related to robust stability and performance can
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
be used. The BIBO stability also guarantees stability of FL (G , K ) making possible usage of performance weights with integration property implying non-existence of state space solutions using DGKF formulae (Doyle et al., 1989) due to zero eigenvalues of appropriate Hamiltonian matrices. Such methodology results in zero steady-state error in the feedback loop with the controller obtained as a solution to equation (13). This technique is neither possible in the scope of the standard μ-synthesis using DGKF formulae, nor using LMI approach (Gahinet and Apkarian, 1994) leading to numerical problems in most of real-world applications. r
e -
Nk Dk
u d
B A
Fig. 4.Schematic diagram of the two-tank system.
The aim of synthesis is to design a controller which satisfies the condition: sup
P Δ [FL (G, K )(Z, D1, , D n n1 n2 , t1, , tn2 )] d 1 ,
ω(–∞,+∞) (15)
where n + n1 + n2 is the order of the nominal feedback system, n1 is the order of particular solution K0, ti are t 0 t1 s t n2 s n2 arbitrary parameters in T and µ' (D n1 1 s ) (D n1 n2 s )
denotes the structured singular value of LFT on generalized plant G and controller K with ªΔ Δ { « del ¬ 0
0º » ΔF ¼
The two-tank system is a benchmark problem showing the efficiency of the controller design. The detailed description is in Balas et al. (1998) and other papers (e.g. Smith et al., 1987; Smith and Doyle, 1988, 1993). Here only points important for the proposed method are presented.
y
Fig. 3. Nominal feedback loop.
Z K stabilizing G
4. TWO-TANK SYSTEM
(16)
where 'del denotes the perturbation matrix and 'F is a fullblock matrix corresponding with the robust performance condition. Tuning parameters are positive and constrained to the real axis since parameters of the transfer function have to be real and due to the fact that non-real poles cause oscillations of the nominal feedback loop. A crucial problem of the cost function in (15) is the fact that many local extremes are present. Hence, local optimization does not yield a suitable or even stabilizing solution. This can be overcome via evolutionary optimization, which solves the task very efficiently. Remark: The proposed method is not the only way of synthesis through pole placement of the nominal feedback loop. Another procedure is possible using pole placement in state space or via the algebraic theory in a general (nondecoupled) sense. These techniques involve purely statespace description or general transfer matrix manipulations and, in the general case, treat both decoupled and nondecoupled feedback systems. In either case, usage of an algorithm of global optimization is necessary due to the multimodality of cost function (15).
The system consists of two water tanks in the cascade depicted in Fig. 4. 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 an overflow. A cold water bias stream also feeds tank 2 and enables the tanks to have different steady-state temperatures. 4.1 Model Description The two-tank system has 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. The third measured signal is the water level in tank 1 (h1), which is not controlled. This quantity is, however, important for the assessment of the controller performance. The nominal model from the inputs fh, fc to the outputs t1, t2 can be written in the two-dimensional transfer matrix ª P ( s) P12 ( s) º Pnom ( s) { « 11 »{ ¬ P21 ( s) P22 ( s)¼
ª 0.0036s 2 0.0001s 7.8157 10 -7 « 3 2 -6 { « s 0.04912s 0.0007s 3.0684 10 -6 « - 0.0109s - 0.0004s 2.3447 10 «¬ s 3 0.0491s 2 0.0007s 3.0684 10 -6
º 0.0004s 4.6643 10 -6 » s 3 0.0491s 2 0.0007s 3.0684 10 -6 » -4 -4 0.1562 10 s 0.0187 10 » 3 2 -6 » s 0.0491s 0.0007s 3.0684 10 ¼
(17)
Differences from the nominal model are treated by multiplicative perturbations at the outputs of the measured quantities (Fig. 5).
Fig. 5. Schematic representation of the perturbed, linear, two-tank model.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
(19)
The majority of the water flowing into tank 2 comes from tank 1, which means that changes in t2 are dominated by changes in t1. It makes more sense to express the reference weighting in terms of t1 and t2 – t1. This allows us to express the fact that t2 is normally commanded to a value close to t1. This is done by using the following weighting approach.
(20)
(25)
The perturbation weights are transfer functions of the form: 0.5s 0.25s 1 20hˆ s
W h1
0.01
W t1
0.1
Wt 2
(18)
1
0.2s 1 100s 0.1 s 21
where hˆ1 0.75 is the steady-state value of h1. For details on the nominal models of tank 1 and 2 see Balas et al. (1998). Sensor noise is modeled by adding unknown weighted input to h1, t1, and t2. The appropriate weights are
Wh1noise = 0.01 Wt1noise = 0.03 Wt2noise = 0.03
(21) (22) (23)
The problem considered is the control of t1 and t2 for command response. The desired closed-loop configuration is illustrated in Fig. 6. The controller has access to both the reference inputs and the temperature measurements. This configuration is less restrictive than the design using only the error between the temperature and the set points. f hc
Tank System
f cc
In this case Wt1cmd
0.1 , Wt diffcmd
0.01
(26)
4.3 Algebraic Approach
K ( s)
K ( s )I 2 det[Pnom ( s )]
1 [Pnom ( s )]1 P11 ( s )
(27)
The choice of decoupling matrix prevents the controller from canceling any poles or zeroes in the right half-plane so that internal stability of the nominal feedback 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
t1cmd t 2 cmd
K
0 º ª w11 º Wtdiffcmd »» «¬ w12 »¼ ¼
0 º ªWt1cmd « 1 »¼ ¬« 0
ª1 «1 ¬
For the purposes of the algebraic μ-synthesis, the MIMO system is decoupled into two identical SISO plants. For decoupling the nominal plant Pnom it is satisfactory to have the controller in the form
4.2 Design Using D-K Iteration
h1 t1 t2
ª t1cmd º «t » ¬ 2cmd ¼
1 det[Pnom ( s )][Pnom ( s )]1 Pnom ( s ) P11 ( s ) 1 det[Pnom ( s )]I 2 P11 ( s )
Pdec ( s )
Let
1 det[Pnom ( s )] P11 ( s )
(28)
Fig. 6. Closed-loop interconnection for D-K iteration.
Pdec {
The layout of the interconnection structure for the D-K iteration is in Fig. 7.
Then the transfer function Pdec can be approximated by a 2nd order system
ª t1cmd º «t » ¬ 2cmd ¼
reference weights
v e1
error weights ª t1meas º « » ¬t 2meas ¼
'
noise weights
+ e2
actuator model
ª f hc º «f » ¬ cc ¼
With respect to degrees of a and b, the transfer functions A, B, DK and NK are defined, so that the number of closed-loop poles is minimal and the asymptotic tracking for step reference signal is achieved:
w2
actuator weights
A
Fig. 7. Interconnection structure for D-K iteration.
40 , Wt 2 perf 400s 1
20 800s 1
a
(s D 2
i 1
The error weights are first-order low-pass filters with Wt1perƒ weighting the t1 tracking error and Wt2perƒ weighting the t2 tracking error. These are given by Wt1 perf
(24)
(30)
and the controller K = NK/DK is obtained by solving the Diophantine equation (13).
z
pert. model
+
0.0027s 2 - 0.0017s 0.0001 s 2 0.0164s 0.0001
b(s) a( s )
Pdec (s )
w1
(29)
DK
i
)
s dK
, B
(s D i ) 4
i 3
b
(s D 2
i 1
, NK
i
)
nK
(s D i ) 4
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(32)
i 3
Degrees of polynomials dK and nK are: ∂dK = 1, ∂nK = 2
(31)
(33)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Thus the characteristic polynomial of the nominal closed loop has 4 pairs of poles –αi, which represent the tuning parameters. The resulting controller K has the structure: K ( s)
nK sd K
(34)
steady-state error is present. The algebraic approach is approx. 7 times faster with no steady-state error. Moreover, the response for the algebraic approach is monotonous for both the measurements and the actuator signals. P
The open-loop interconnection differs from the D-K iteration. Performance weights are as follows: Wt1 perf Wt1cmd
130s 1 , W t 2 perf 400s 1 1 , Wt diffcmd 1
130s 0.5 800s 1
0.8 0.7
0.5 0.4
(36)
0.3 0.2
The input to the controller is the tracking error of the measured temperatures t1err, t2err (see Fig. 8). As in the previous model the relaxed performance weight is justified by the additional postulate of decoupled control for the nominal closed loop, which is not present in the D-K iteration case. The controller design is reduced to minimization of the peak of μ-function. The cost function is then defined as (37)
reference weights
v e1
error weights
ª t1err º «t » ¬ 2err ¼
noise weights
+
e2
0.1 0 -10 10
ª f hc º «f » ¬ cc ¼
w2
actuator weights
Fig. 8. Interconnection structure for the algebraic approach.
-8
10
-6
-4
10 Frequency (rad/s)
10
-2
10
0
10
2
0.8
0.9
0.75
0.8
0.7
fhc fhc fcc fcc
0.7
h1 h1 t1 t1 t2 t2
0.65
0.6
0.6
0.5
0.55
0.4
0.5
0.3
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 10. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the D-K iteration and nominal plant.
z
actuator model
10
Fig. 9. Comparison of μ-plots.
0.45
pert. model
+
+
w1 '
Algebraic approach Algebraic approach D-K iteration iteration D-K
0.6
(35)
||P'[FL(G, K)]||f
1 0.9
0.9
0.75
0.8
0.7
h1 h1 t1 t1 t2 t2
0.65
G
0.6
0.6
0.5
0.55
0.4
0.5
Differential Migration (DM, Dlapa, 2009) was used for searching of the optimal values of αi. Rough results obtained from DM were then tuned up by the Nelder-Mead simplex method. The poles were constrained to the intervals of 0 to –300. 4.4 Simulation Studies In order to assess performance of the resulting controllers, simulations of response to a ramp reference signal were done. Through the D-K iteration, a controller with 42 states was obtained. The algebraic approach yields a controller with 14 states, which guarantees decoupled control for the nominal plant. Robust stability and performance can be observed by μ-plots in Fig. 9. The μ-plots for both controllers are lower than 1, which implies that the robust stability has been achieved. The simulation of response to the ramp reference signal for the D-K iteration and the nominal plant is in Fig. 10. The simulation for the nominal plant with the same set point and algebraic approach is in Fig. 11. It follows from the graphs that the D-K iteration has slower response and a
0.45
fhc fhc fcc fcc
0.7
0.3
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 11. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the algebraic approach and nominal plant. 0.8
0.9
0.75
0.8
0.7 0.65 0.6
0.6 0.5
0.55
0.4
0.5
0.3
0.45
fhc fhc fcc fcc
0.7
h1 h1 t1 t1 t2 t2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 12. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the D-K iteration and perturbed plant.
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0.75
0.9
0.7
0.8
h1 t1 t1 t2 t2
0.65
searching the decoupling model. The D-K iteration without decoupling makes a trade-off between robust stability and performance. Therefore, higher robustness is reached as it fully utilizes the MIMO structure of the controller. However, the higher stability is achieved at the expense of worse performance.
fhc fhc fcc fcc
0.7 0.6
0.6
0.5 0.55
0.45
Although generally the presented method cannot substitute the D-K iteration, in some cases, the algebraic approach yields simple controller structure and reduces the complexity of its derivation. Besides the simpler procedure, usage of performance weights with integration property is possible. This is not true for DGKF formulae. Although LMI approach gives a solution to this problem, in practice, numerical difficulties arise causing impossibility of controller computation. The problem can be overcome using IMC design with integrator cascade but with higher complexity of the resulting controller than for the proposed technique.
0.4
0.5
0.3 0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 13.Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the algebraic approach and perturbed plant. 0.9
0.95 0.9
h1 h1
0.8
t1 t1
0.85
t2 t2
0.8 0.75
0.6
0.7
This work was supported by the Ministry of Education Youth and Sports of the Czech Republic under Grant MSM7088352102 and by the European Regional Development Fund under the Project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089.
0.5
0.65 0.6
0.4
0.55
0.3
0.5 0.45
ACKNOWLEDGMENTS
fhc fhc fcc fcc
0.7
0
500
1000
1500
Time (seconds)
2000
2500
0.2
0
500
1000
1500
Time (seconds)
2000
REFERENCES
2500
Fig. 14. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and (from 1000 to 1020 seconds) t2 from 0.67 to 0.47 for the algebraic approach and nominal plant. 0.95
0.9
0.9
h1 h1 t1 t1 t2 t2
0.85 0.8
0.8
0.6
0.7
0.5
0.65 0.6
0.4
0.55
0.3
0.5 0.45
fhc fhc fcc fcc
0.7
0.75
0
500
1000
1500
Time (seconds)
2000
2500
0.2
0
500
1000
1500
Time (seconds)
2000
2500
Fig. 15.Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and (from 1000 to 1020 seconds) t2 (dot-dash) from 0.67 to 0.47 for the algebraic approach and perturbed plant.
Although simulations in Fig. 14 and 15 show that the decoupling property is held for the nominal plant, under perturbation the feedback system is only almost-decoupled since real decoupling is guaranteed solely for the nominal case. 5. CONCLUSION
An application of the algebraic approach to a MIMO system has been presented. The plant was decoupled into two identical SISO systems and controller was designed via optimization of the poles of the nominal feedback loop. The performance and robustness were evaluated by Hf norm of the μ-function. The resulting controller has simple structure and satisfies robust stability and performance condition. The simulation proved that the algebraic approach has monotonous step response and significantly faster set point tracking for the ramp and stepwise reference signal than the D-K iteration. Moreover, the asymptotic tracking is achieved, which is not true for 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 and by the choice of the controller structure (PI). This scheme can be used in the scope of the standard tool using model matching (see Prempain and Bergeron, 1998) with an additional step of
Balas, G. J., Doyle, J. C., Glover, K., Packard, A. and Smith, R. (1998). µ-Analysis and Synthesis Toolbox for Use with MATLAB. The MathWorks, Inc. Dlapa, M. (2009). “Differential Migration: Sensitivity Analysis and Comparison Study,” In 2009 IEEE Congress on Evolutionary Computation (IEEE CEC 2009), pp. 17291736, ISBN 978-1-4244-2959-2. Dlapa, M., Prokop, R. and Bakošová, M. (2009). “Robust Control of a Two Tank System Using Algebraic Approach,” EUROCAST 2009, pp. 603-609, ISBN 978-84-691-8502-5. Dlapa, M. and Prokop, R. (2010): Algebraic approach to controller design using structured singular value, Control Engineering Practice, Vol. 18, No. 4, pp. 358–382, ISSN 0967-0661. Doyle, J.C. (1982). “Analysis of feedback systems with structured uncertainties,” In Proceedings of IEEE, Part-D, 129, pp. 242250. Doyle, J.C. (1985). “Structured uncertainty in control system design,” In Proceedings of 24th IEEE Conference on Decision and Control, pp. 260-265. Doyle, J. C., Glover, K., Khargonekar, P. P. and Francis, B. A. (1989). State-Space Solutions to Standard H2 and H∞ Control Problems. IEEE Transactions on Automatic Control, 34(8), 831-847. Gahinet, P. and Apkarian, P. (1994). A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control, 4, 421-449. Kučera, V. (1972). Discrete Linear Control: The Polynomial Equation Approach. Wiley, New York. Kučera, V. (1993). Diophantine equations in control-A survey. Automatica, 29(6), 1361-75. Prempain, E. and Bergeron, B (1998). A multivariable twodegree-of-freedom control methodology. Automatica, Elsevier Science Ltd., 34(12), 1601-1606. Smith, R. S., Doyle, J., Morari, M. and Skjellum, A. (1987). A case study using P: Laboratory process control problem. Proceedings Int. Fed. Auto. Control, 8, 403–415. Smith, R. S. and Doyle, J. (1988). The two tank experiment: A benchmark control problem. Proc. Amer. Control Conf., 3, 403–415, 1988. Smith, R. S. and Doyle, J. C. (1993). Closed loop relay estimation of uncertainty bounds for robust control models. In Proceedings of the 12th IFAC World Congress, Vol. 9, pp. 57–60. Stein, G. and Doyle, J. C. (1991). Beyond singular values and loopshapes/ AIAA Journal of Guidance and Control, 14(1), 5-16. Vidyasagar, M. (1985). Control systems synthesis: a factorization approach, MIT Press. Cambridge, MA.
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Controller Design for a Two Tank System Using Structured Singular Value and Direct Search Methods M. Dlapa* *Tomas Bata University in Zlin, Faculty of Applied Informatics, nám. T.G.Masaryka 5555, 760 01 Zlín Czech Republic (e-mail: [email protected]; Fax: +420 57 603 5279). Abstract: The aim of this paper is to show an algebraic approach to controller design using the structured singular value (denoted μ) as a robust stability and performance indicator. The algebraic μ-synthesis is applied to a two-tank system, which is a well known benchmark problem. The design is performed using decoupling of the nominal plant into two identical SISO (single-input single-output) systems. Global optimization methods are used due to multimodality of the cost function. The overall performance is verified by simulations for the nominal and perturbed system and compared with the D-K iteration. Keywords: Robust control; evolutionary computation; algebraic approaches; structured singular value; process control. 1. INTRODUCTION The algebraic theory (e.g. Kučera, 1972, 1993; Vidyasagar, 1985) is well known and its importance is growing due to the simplicity of controller derivation and the fact that some crucial properties of the resulting feedback loop can be easily influenced by the choice of the controller structure, which is not hard to do within the scope of this approach. The structured singular value denoted µ (see Doyle, 1982, 1985) provides a measure of robust stability and performance that can take into account many aspects of controller design including sensor noise, dynamic perturbations as well as parametric uncertainties if the linear fractional transformation (LFT) can be used for its treatment. Standard tools for µ-synthesis provide a methodology for controller design with decoupling using model matching (see Prempain and Bergeron, 1998). However, the need of defining the internal model brings another step into the process of obtaining the controller. The algebraic approach provides the methodology for synthesis of very simple controllers (PI, PID) without an additional model matching design, which simplifies controller derivation. Due to the multimodality of the cost function, an algorithm for global optimization is used for tuning nominal closed-loop pole placement, where the peak of the µ-function in frequency domain gives the measure of controller stability and performance. Knowing this, behavior of the resulting feedback system can be simulated for the worst-case perturbation causing the highest value of µ. Algebraic methods are easy to apply to SISO (single-input single-output) systems described by continuous or discrete transfer functions. Still, if used for MIMO (multi-input multioutput) systems, computational difficulties are severe. In this paper, the problem of the MIMO system design is solved via decoupling a MIMO system into two identical SISO plants, 978-3-902661-93-7/11/$20.00 © 2011 IFAC
which are then approximated by transfer functions with a simple structure. This guarantees decoupled control and simplifies derivation of the pole placement formulae. The algebraic μ-synthesis (Dlapa et al., 2009; Dlapa and Prokop, 2010) overcomes some difficulties connected with the D-K iteration, namely the fact that it does not guarantee convergence to a global or even local minimum (Stein and Doyle, 1991). Controllers obtained via the algebraic approach can have simpler structure due to the fact that there is no need of scaling matrices absorbance into the generalized plant, and hence no need of further simplification causing deterioration of the frequency properties of the resulting controller. Moreover, the controller structure can be chosen in advance, which is not possible in the scope of currently used methods. In this paper, the algebraic μ-synthesis is applied to the control of a two-tank system - classical example of robust control design. The D-K iteration is used as a reference method and the results are compared through simulations for the nominal and perturbed plant.
The following notation is used: || ||f denotes Hf norm, R and Cnum are real numbers and complex matrices, respectively, In is the unit matrix of dimension n and RPS denotes the ring of Hurwitz-stable and proper rational functions. 2. PRELIMINARIES
Define ' as a set of block diagonal matrices
Δ { {diag[G 1 I r1 , , G S I rS , '1 , , ' F ] : G i C, ' j C
where
m j um j
} (1)
S is the number of repeated scalar blocks, F is the number of full blocks, r1,}, rS and m1,}, mF are positive integers defining dimensions of scalar and full blocks.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
For consistency among all the dimensions, the following condition must be held
¦r ¦m S
i 1
F
i
j 1
j
n
(2)
Definition 1: For M C
nun
is μ'(M) defined as
1 P Δ (M ) { min{V (' ) : ' Δ, det(I M' )
0}
SISO plants. The nominal model is defined in terms of transfer functions: ª P11 ( s ) P1l ( s ) º Pnom ( s ) { «« »» ¬« Pl1 ( s ) Pll ( s ) ¼»
(7)
For decoupling the nominal plant Pnom (Pnom invertible) it is satisfactory to have the controller in the form
(3)
(8)
K ( s)
Consider a complex matrix M partitioned as
where Pxy is an element of adj[Pnom(s)] = det[Pnom(s)][Pnom(s)]–1 with the highest degree of numerator {adj[Pnom(s)] denotes adjugate matrix of Pnom}. The choice of the decoupling matrix prevents the controller from cancelling any poles or zeros from the right half-plane so that internal stability of the nominal feedback loop is held. The MIMO problem is reduced to finding a controller K(s), which is tuned via setting the poles of the nominal feedback loop with decoupled plant
M
ª M11 M12 º « » ¬M 21 M 22 ¼
(4)
and suppose there is a defined block structure '2 which is compatible in size with M22 (for any '2 '2, M22'2 is square). For '2 '2, consider the following loop equations e = M11d + M12w z = M21d + M22w (5) w = ' 2z If the inverse to I – M22'2 exists, then e and d must satisfy e = FL(M, '2)d, where
FL(M, '2) = M11 + M12'2(I – M22'2) M21 (6) is a linear fractional transformation on M by '2, and in a feedback diagram appears as the loop in Fig. 1. –1
The subscript L on FL pertains to the lower loop of M and is closed by '2. An analogous formula describes FU(M, '1), which is the resulting matrix obtained by closing the upper loop of M with a matrix '1 '1. e z
Define Pdec {
1 det[Pnom ( s )] Pxy ( s )
Pdec (s )
* Pdec (s)
Fig. 1. LFT interconnection. 3. ALGEBRAIC μ-SYNTHESIS The algebraic μ-synthesis can be applied to any control problem that can be transformed to the loop in Fig. 2, where G denotes the generalized plant, K is the controller, 'del is the perturbation matrix, r is the reference and e is the output.
errors
(10)
b( s ) a(s)
(11)
b0 b1s bn s n (D 1 s)(D 2 s) (D n s) s n a0 a1 s an1 s n1 (D 1 s)(D 2 s) (D n s)
B , A, B RPS A
(12)
The controller K = NK/DK is obtained by solving the Diophantine equation
ADK + BNK = 1 (13) with A, B, DK, NK Rps. Equation (13) is often called the Bezout identity. All feedback controllers NK/DK are given by K
Δdel
G
(9)
* Transfer function Pdec can be approximated by a system Pdec with lower order than Pdec
w '
e
1 det[Pnom ( s )][Pnom ( s )]1 Pnom ( s ) Pxy ( s ) 1 det[Pnom ( s )]I l Pxy ( s )
Pdec ( s )
which can be rewritten in terms of its coefficients and transformed to the elements of RPS
d
M
K ( s)I l det[Pnom ( s)]
1 [Pnom ( s )]1 Pxy ( s )
If no such ' ' exists making I – M' singular, then μ'(M) = 0.
NK DK
N K 0 AT DK 0 BT
,
N K 0 , DK 0 R PS
(14)
where N K 0 , DK 0 R PS are particular solutions of (13) and T is
r
an arbitrary element of RPS.
controls K
Fig. 2. Closed loop interconnection. For the purposes of the algebraic μ-synthesis, the MIMO system with l inputs and l outputs has to be decoupled into l identical
The controller K satisfying equation (13) guarantees the BIBO (bounded input bounded output) stability of the feedback loop in Fig. 3. This is a crucial point for the theorems regarding the structured singular value. If the BIBO stability is held, then the nominal model is internally stable and theorems related to robust stability and performance can
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
be used. The BIBO stability also guarantees stability of FL (G , K ) making possible usage of performance weights with integration property implying non-existence of state space solutions using DGKF formulae (Doyle et al., 1989) due to zero eigenvalues of appropriate Hamiltonian matrices. Such methodology results in zero steady-state error in the feedback loop with the controller obtained as a solution to equation (13). This technique is neither possible in the scope of the standard μ-synthesis using DGKF formulae, nor using LMI approach (Gahinet and Apkarian, 1994) leading to numerical problems in most of real-world applications. r
e -
Nk Dk
u d
B A
Fig. 4.Schematic diagram of the two-tank system.
The aim of synthesis is to design a controller which satisfies the condition: sup
P Δ [FL (G, K )(Z, D1, , D n n1 n2 , t1, , tn2 )] d 1 ,
ω(–∞,+∞) (15)
where n + n1 + n2 is the order of the nominal feedback system, n1 is the order of particular solution K0, ti are t 0 t1 s t n2 s n2 arbitrary parameters in T and µ' (D n1 1 s ) (D n1 n2 s )
denotes the structured singular value of LFT on generalized plant G and controller K with ªΔ Δ { « del ¬ 0
0º » ΔF ¼
The two-tank system is a benchmark problem showing the efficiency of the controller design. The detailed description is in Balas et al. (1998) and other papers (e.g. Smith et al., 1987; Smith and Doyle, 1988, 1993). Here only points important for the proposed method are presented.
y
Fig. 3. Nominal feedback loop.
Z K stabilizing G
4. TWO-TANK SYSTEM
(16)
where 'del denotes the perturbation matrix and 'F is a fullblock matrix corresponding with the robust performance condition. Tuning parameters are positive and constrained to the real axis since parameters of the transfer function have to be real and due to the fact that non-real poles cause oscillations of the nominal feedback loop. A crucial problem of the cost function in (15) is the fact that many local extremes are present. Hence, local optimization does not yield a suitable or even stabilizing solution. This can be overcome via evolutionary optimization, which solves the task very efficiently. Remark: The proposed method is not the only way of synthesis through pole placement of the nominal feedback loop. Another procedure is possible using pole placement in state space or via the algebraic theory in a general (nondecoupled) sense. These techniques involve purely statespace description or general transfer matrix manipulations and, in the general case, treat both decoupled and nondecoupled feedback systems. In either case, usage of an algorithm of global optimization is necessary due to the multimodality of cost function (15).
The system consists of two water tanks in the cascade depicted in Fig. 4. 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 an overflow. A cold water bias stream also feeds tank 2 and enables the tanks to have different steady-state temperatures. 4.1 Model Description The two-tank system has 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. The third measured signal is the water level in tank 1 (h1), which is not controlled. This quantity is, however, important for the assessment of the controller performance. The nominal model from the inputs fh, fc to the outputs t1, t2 can be written in the two-dimensional transfer matrix ª P ( s) P12 ( s) º Pnom ( s) { « 11 »{ ¬ P21 ( s) P22 ( s)¼
ª 0.0036s 2 0.0001s 7.8157 10 -7 « 3 2 -6 { « s 0.04912s 0.0007s 3.0684 10 -6 « - 0.0109s - 0.0004s 2.3447 10 «¬ s 3 0.0491s 2 0.0007s 3.0684 10 -6
º 0.0004s 4.6643 10 -6 » s 3 0.0491s 2 0.0007s 3.0684 10 -6 » -4 -4 0.1562 10 s 0.0187 10 » 3 2 -6 » s 0.0491s 0.0007s 3.0684 10 ¼
(17)
Differences from the nominal model are treated by multiplicative perturbations at the outputs of the measured quantities (Fig. 5).
Fig. 5. Schematic representation of the perturbed, linear, two-tank model.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
(19)
The majority of the water flowing into tank 2 comes from tank 1, which means that changes in t2 are dominated by changes in t1. It makes more sense to express the reference weighting in terms of t1 and t2 – t1. This allows us to express the fact that t2 is normally commanded to a value close to t1. This is done by using the following weighting approach.
(20)
(25)
The perturbation weights are transfer functions of the form: 0.5s 0.25s 1 20hˆ s
W h1
0.01
W t1
0.1
Wt 2
(18)
1
0.2s 1 100s 0.1 s 21
where hˆ1 0.75 is the steady-state value of h1. For details on the nominal models of tank 1 and 2 see Balas et al. (1998). Sensor noise is modeled by adding unknown weighted input to h1, t1, and t2. The appropriate weights are
Wh1noise = 0.01 Wt1noise = 0.03 Wt2noise = 0.03
(21) (22) (23)
The problem considered is the control of t1 and t2 for command response. The desired closed-loop configuration is illustrated in Fig. 6. The controller has access to both the reference inputs and the temperature measurements. This configuration is less restrictive than the design using only the error between the temperature and the set points. f hc
Tank System
f cc
In this case Wt1cmd
0.1 , Wt diffcmd
0.01
(26)
4.3 Algebraic Approach
K ( s)
K ( s )I 2 det[Pnom ( s )]
1 [Pnom ( s )]1 P11 ( s )
(27)
The choice of decoupling matrix prevents the controller from canceling any poles or zeroes in the right half-plane so that internal stability of the nominal feedback 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
t1cmd t 2 cmd
K
0 º ª w11 º Wtdiffcmd »» «¬ w12 »¼ ¼
0 º ªWt1cmd « 1 »¼ ¬« 0
ª1 «1 ¬
For the purposes of the algebraic μ-synthesis, the MIMO system is decoupled into two identical SISO plants. For decoupling the nominal plant Pnom it is satisfactory to have the controller in the form
4.2 Design Using D-K Iteration
h1 t1 t2
ª t1cmd º «t » ¬ 2cmd ¼
1 det[Pnom ( s )][Pnom ( s )]1 Pnom ( s ) P11 ( s ) 1 det[Pnom ( s )]I 2 P11 ( s )
Pdec ( s )
Let
1 det[Pnom ( s )] P11 ( s )
(28)
Fig. 6. Closed-loop interconnection for D-K iteration.
Pdec {
The layout of the interconnection structure for the D-K iteration is in Fig. 7.
Then the transfer function Pdec can be approximated by a 2nd order system
ª t1cmd º «t » ¬ 2cmd ¼
reference weights
v e1
error weights ª t1meas º « » ¬t 2meas ¼
'
noise weights
+ e2
actuator model
ª f hc º «f » ¬ cc ¼
With respect to degrees of a and b, the transfer functions A, B, DK and NK are defined, so that the number of closed-loop poles is minimal and the asymptotic tracking for step reference signal is achieved:
w2
actuator weights
A
Fig. 7. Interconnection structure for D-K iteration.
40 , Wt 2 perf 400s 1
20 800s 1
a
(s D 2
i 1
The error weights are first-order low-pass filters with Wt1perƒ weighting the t1 tracking error and Wt2perƒ weighting the t2 tracking error. These are given by Wt1 perf
(24)
(30)
and the controller K = NK/DK is obtained by solving the Diophantine equation (13).
z
pert. model
+
0.0027s 2 - 0.0017s 0.0001 s 2 0.0164s 0.0001
b(s) a( s )
Pdec (s )
w1
(29)
DK
i
)
s dK
, B
(s D i ) 4
i 3
b
(s D 2
i 1
, NK
i
)
nK
(s D i ) 4
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(32)
i 3
Degrees of polynomials dK and nK are: ∂dK = 1, ∂nK = 2
(31)
(33)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Thus the characteristic polynomial of the nominal closed loop has 4 pairs of poles –αi, which represent the tuning parameters. The resulting controller K has the structure: K ( s)
nK sd K
(34)
steady-state error is present. The algebraic approach is approx. 7 times faster with no steady-state error. Moreover, the response for the algebraic approach is monotonous for both the measurements and the actuator signals. P
The open-loop interconnection differs from the D-K iteration. Performance weights are as follows: Wt1 perf Wt1cmd
130s 1 , W t 2 perf 400s 1 1 , Wt diffcmd 1
130s 0.5 800s 1
0.8 0.7
0.5 0.4
(36)
0.3 0.2
The input to the controller is the tracking error of the measured temperatures t1err, t2err (see Fig. 8). As in the previous model the relaxed performance weight is justified by the additional postulate of decoupled control for the nominal closed loop, which is not present in the D-K iteration case. The controller design is reduced to minimization of the peak of μ-function. The cost function is then defined as (37)
reference weights
v e1
error weights
ª t1err º «t » ¬ 2err ¼
noise weights
+
e2
0.1 0 -10 10
ª f hc º «f » ¬ cc ¼
w2
actuator weights
Fig. 8. Interconnection structure for the algebraic approach.
-8
10
-6
-4
10 Frequency (rad/s)
10
-2
10
0
10
2
0.8
0.9
0.75
0.8
0.7
fhc fhc fcc fcc
0.7
h1 h1 t1 t1 t2 t2
0.65
0.6
0.6
0.5
0.55
0.4
0.5
0.3
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 10. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the D-K iteration and nominal plant.
z
actuator model
10
Fig. 9. Comparison of μ-plots.
0.45
pert. model
+
+
w1 '
Algebraic approach Algebraic approach D-K iteration iteration D-K
0.6
(35)
||P'[FL(G, K)]||f
1 0.9
0.9
0.75
0.8
0.7
h1 h1 t1 t1 t2 t2
0.65
G
0.6
0.6
0.5
0.55
0.4
0.5
Differential Migration (DM, Dlapa, 2009) was used for searching of the optimal values of αi. Rough results obtained from DM were then tuned up by the Nelder-Mead simplex method. The poles were constrained to the intervals of 0 to –300. 4.4 Simulation Studies In order to assess performance of the resulting controllers, simulations of response to a ramp reference signal were done. Through the D-K iteration, a controller with 42 states was obtained. The algebraic approach yields a controller with 14 states, which guarantees decoupled control for the nominal plant. Robust stability and performance can be observed by μ-plots in Fig. 9. The μ-plots for both controllers are lower than 1, which implies that the robust stability has been achieved. The simulation of response to the ramp reference signal for the D-K iteration and the nominal plant is in Fig. 10. The simulation for the nominal plant with the same set point and algebraic approach is in Fig. 11. It follows from the graphs that the D-K iteration has slower response and a
0.45
fhc fhc fcc fcc
0.7
0.3
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 11. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the algebraic approach and nominal plant. 0.8
0.9
0.75
0.8
0.7 0.65 0.6
0.6 0.5
0.55
0.4
0.5
0.3
0.45
fhc fhc fcc fcc
0.7
h1 h1 t1 t1 t2 t2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 12. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the D-K iteration and perturbed plant.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
0.75
0.9
0.7
0.8
h1 t1 t1 t2 t2
0.65
searching the decoupling model. The D-K iteration without decoupling makes a trade-off between robust stability and performance. Therefore, higher robustness is reached as it fully utilizes the MIMO structure of the controller. However, the higher stability is achieved at the expense of worse performance.
fhc fhc fcc fcc
0.7 0.6
0.6
0.5 0.55
0.45
Although generally the presented method cannot substitute the D-K iteration, in some cases, the algebraic approach yields simple controller structure and reduces the complexity of its derivation. Besides the simpler procedure, usage of performance weights with integration property is possible. This is not true for DGKF formulae. Although LMI approach gives a solution to this problem, in practice, numerical difficulties arise causing impossibility of controller computation. The problem can be overcome using IMC design with integrator cascade but with higher complexity of the resulting controller than for the proposed technique.
0.4
0.5
0.3 0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
1200
Time (seconds)
1400
1600
1800
2000
Fig. 13.Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and t2 (dot-dash) from 0.67 to 0.47 for the algebraic approach and perturbed plant. 0.9
0.95 0.9
h1 h1
0.8
t1 t1
0.85
t2 t2
0.8 0.75
0.6
0.7
This work was supported by the Ministry of Education Youth and Sports of the Czech Republic under Grant MSM7088352102 and by the European Regional Development Fund under the Project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089.
0.5
0.65 0.6
0.4
0.55
0.3
0.5 0.45
ACKNOWLEDGMENTS
fhc fhc fcc fcc
0.7
0
500
1000
1500
Time (seconds)
2000
2500
0.2
0
500
1000
1500
Time (seconds)
2000
REFERENCES
2500
Fig. 14. Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and (from 1000 to 1020 seconds) t2 from 0.67 to 0.47 for the algebraic approach and nominal plant. 0.95
0.9
0.9
h1 h1 t1 t1 t2 t2
0.85 0.8
0.8
0.6
0.7
0.5
0.65 0.6
0.4
0.55
0.3
0.5 0.45
fhc fhc fcc fcc
0.7
0.75
0
500
1000
1500
Time (seconds)
2000
2500
0.2
0
500
1000
1500
Time (seconds)
2000
2500
Fig. 15.Response to reference signal which ramps (from 80 to 100 seconds) t1 (solid) from 0.75 to 0.57, and (from 1000 to 1020 seconds) t2 (dot-dash) from 0.67 to 0.47 for the algebraic approach and perturbed plant.
Although simulations in Fig. 14 and 15 show that the decoupling property is held for the nominal plant, under perturbation the feedback system is only almost-decoupled since real decoupling is guaranteed solely for the nominal case. 5. CONCLUSION
An application of the algebraic approach to a MIMO system has been presented. The plant was decoupled into two identical SISO systems and controller was designed via optimization of the poles of the nominal feedback loop. The performance and robustness were evaluated by Hf norm of the μ-function. The resulting controller has simple structure and satisfies robust stability and performance condition. The simulation proved that the algebraic approach has monotonous step response and significantly faster set point tracking for the ramp and stepwise reference signal than the D-K iteration. Moreover, the asymptotic tracking is achieved, which is not true for 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 and by the choice of the controller structure (PI). This scheme can be used in the scope of the standard tool using model matching (see Prempain and Bergeron, 1998) with an additional step of
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