A New Scheme on Robust Observer Based Control Design for

Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11­13, 2007

FrB20.4

A New Scheme on Robust Observer Based Control Design for Nonlinear Interconnected Systems with Application to an Industrial Utility Boiler Adarsha Swarnakar, Horacio Jose Marquez and Tongwen Chen Abstract— This paper presents a new design algorithm for the decentralized output feedback control problem of largescale interconnected systems. A decentralized estimated state feedback controller and a decentralized observer are designed for each subsystem, based on linear matrix inequalities (LMIs). Sufficient conditions for the synthesis of feedback action are provided, under which the proposed controllers and observers can achieve robust stabilization of the overall large-scale system. An attractive feature of the proposed scheme is that it guarantees connective stability of the overall system, and requires no intersubsystem communication. The controller design is evaluated on a natural circulation drum boiler, where the nonlinear model describes the key dynamical properties of the drum, the risers, the downcomers and the turbine-generator unit. The linearized system has two poles at origin, one associated with water dynamics and the other with generator dynamics. Simulation results show the effectiveness of the proposed control against instabilities following sudden load variations. The control is also effective for steady state operation.

I. I NTRODUCTION During the last few decades, many researchers in the field of large-scale systems are devoted to decentralized robust control strategies [1], [2], [3], [4], [5], [6]. An important motivation for the design of decentralized schemes is that the interconnected systems can be decomposed into low-order subsystems. Therefore, the design procedure gets simplified and the overall computational effort can be shared by all the subsystem controllers. Also, in diverse fields like power systems, robotics and space structures, interconnected systems do not stay in one piece during operation. In many cases, the subsystems are disconnected and again connected in an unpredictable way to perform programmed tasks [3]. Under such structural reconfigurations, what is required is a control strategy that can guarantee connective stability of the overall system [1], [3], or can achieve a desired robust performance in the presence of uncertain interconnections [2], [7]. The concept of decentralized control is also interesting, because it leads to a reduction in communication overhead. It has been long recognized that the attractive property of decentralized information structure disappears when the states of subsystems are not available at the local level. Hence, there are strong research efforts to build decentralized feedback controllers based on measurements only [8], [9], or design decentralized observers to estimate the state of individual subsystems that can be used for state feedback This work was supported by Natural Sciences and Engineering Research Council of Canada and Syncrude Canada Limited. Adarsha Swarnakar, Horacio J. Marquez, and Tongwen Chen are with Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada. (adarsha, marquez,

tchen)@ece.ualberta.ca

1­4244­0989­6/07/$25.00 ©2007 IEEE.

control [10], [1], [11]. However, these results are based on the fact that, in order to utilize the separation principle in the design of individual observers, it is necessary that all observers exchange their computed state estimates. The design is therefore unattractive, and can be prone to failure in communication links, delays and cost of inter-subsystem communications. Moreover, as most of the ideas are mainly devoted to linear systems, the controller may not guarantee a satisfactory performance over a wide range of operating conditions. The last decade has seen a number of new developments in the design of observer based control schemes for special class of nonlinear systems [12], [7], [3], [5], [6]. It is well known that as the separation principle may not be applicable to nonlinear systems [13], if the true state is replaced by the estimate given by an observer, then exponential stability of the observer does not, in general, guarantee closed loop stability. Considering the challenge of designing decentralized observer based controllers, [3] proposed a method of autonomous decentralized control design, which requires no communications between the subsystems and, at the same time, guarantees connective stability of the overall closed-loop system. The class of nonlinear system covered by this work is also fairly general, and the design utilizes the multifaceted tools of LMI [14]. However, for a solution to exist, this formulation also requires, for each subsystem, that the number of control inputs must be equal to the dimension of the state, which is very restrictive. To resolve this issue, [5] presented some sufficient conditions of observer-based control design. It utilizes the concept of distance to uncontrollability (unobservability) of a pair of matrices (A, B)((C, A)) [15], [16], and is based on the existence of symmetric positive definite solutions to algebraic Riccati equations (AREs). However, it does not maximize the interconnection bounds, which is important in practical fields like power systems because they are always subjected to uncertain structural perturbations. It also requires selection of many additional design parameters for each subsystem, which has to be done on a trial and error basis, and so it may not be very user friendly to engineers. In [6], a sequential two-step design procedure has been introduced to solve a non-convex optimization problem for the nonlinear system in [3], [5]. It is interesting that it provides a way to maximize the interconnection bounds; however, the optimization problem cannot give a global minima because the set of solutions are not convex. In this paper, we propose a new design algorithm for decentralized observer based controllers. We show that the

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FrB20.4 restrictiveness of the design in [3], [5], [6] can be resolved by providing additional degrees of freedom in the observer design, and the existence of decentralized stabilizing controllers and observers depend on the feasibility of solving an optimization problem in the LMI framework. It is important to note that the algorithm provides only sufficient conditions for quadratic stability. Nevertheless, LMI-based design is always preferable because of its simplicity, its ineffable grace of design, its ability to capture a wide panorama of control problems and from a numerical perspective, as it not only computes the controller and observer parameters by guaranteeing exponential stabilization, but it also maximizes the interconnection bounds. It is useful to learn that the proposed LMI formulation recognizes the matching conditions by returning design parameters for any prescribed bound on the interconnection terms. The control law also guarantees connective stability of the overall system. To show that the approach is also practically relevant, the controller design is evaluated on the nonlinear model of a power unit which describes the key dynamical properties of drum, risers, downcomers, and the turbine-generator set. We extend the model of [17] to incorporate the dynamics of governor, turbine, and generator. Thus, interconnection terms between the steam generating unit and the electricity generating unit are identified. This helps us to move the pole associated with the pressure dynamics to the left half plane. Throughout the work, a nonlinear simulation package of boilers called SYNSIM is used, which incorporates the nonlinearities encountered in the true plant. It is developed by Syncrude Canada Limited with the purpose of simulating certain upset conditions that have been sporadically detected as well as a general tool for stability analysis. At present, it is extensively used by the power plant engineers, because the correlation between measurement from the true plant outputs and the predictions by SYNSIM is very good. Our experiments with SYNSIM reveals that the overall system is highly interacting and nonlinear. In addition to it, linearization of the developed nonlinear model gives two poles at the origin: one associated with the water dynamics, and the other with the generator dynamics. Considering these issues, we decomposed the overall system into two subsystems: a) drum-boiler and b) governor, turbine, generator unit. Robust decentralized observer-based controllers are then designed, which can maintain stability in the presence of sudden load variations. The stabilizing effect of controllers are also tested for different initial conditions, and simulation results are provided to show the effectiveness of the proposed approach. II. D ECENTRALIZED O BSERVER - BASED C ONTROLLERS IN THE LMI F RAMEWORK Let us consider a nonlinear process of the form [18], [9]: x(t) ˙ = Ax(t) + Bu(t) + Ghr (t, x), y(t) = Cx(t),

(1)

where x(t) ∈ "n is the state of the system, u(t) ∈ "m is the input vector and y(t) ∈ "p is the output vector. A, B, C, and G are constant n × n, n × m, p × n, and n × n matrices, respectively. The term hr : "n+1 −→ "n is a

piecewise continuous nonlinear function in both arguments t and x, satisfying hr (t, 0) = 0. Assume that (A, B) is stabilizable, (C, A) is detectable and the uncertain term hr (t, x) is bounded by an inequality [18], [4], [9] hTr (t, x)hr (t, x)

≤ α2 xT HrT Hr x,

(2)

where Hr is a known constant matrix and α > 0 is a scalar parameter which can be termed as a degree of robustness. Maximization of α leads to increased robustness of the closed loop system against uncertain nonlinear perturbations. Consider the following open loop dynamic observer x ˆ˙ (t)

=

Aˆ x(t) + Bu(t) + u ˆ, yˆ(t) = C x ˆ(t),

(3)

where x ˆ ∈ "n is the estimated state vector, uˆ ∈ "n is an additional observer input that can be used to achieve closedloop estimation. In our approach, a dynamic linear feedback in the form of u ˆ = Luˆey (y − yˆ) is used, where ˆ), u ˆ = Cl xl + Dl C(x − x ˆ). Luˆey : {x˙ l = Al xl + Bl C(x − x If a static state feedback controller u = K x ˆ is used, then the closed loop system takes the form     A + BK Dl C Cl 0 0 A − Dl C −Cl  z(t) +  G  z(t) ˙ =  0 Bl C Al 0 ˆ ×w(t, z) = Az(t) + Gr w(t, z), (4) % T &T T T ˆ (t) e (t) xl (t) where z(t) = x , the state estimation error e = (x − x ˆ), and the nonlinear function hr (t, x) in terms of z is represented by w(t, z), which satisfies  T  Hr Hr HrT Hr 0 wT (t, z)w(t, z) ≤ α2 z T  HrT Hr HrT Hr 0  z 0 0 0 =

α2 z T (t)H T Hz(t).

(5)

Considering power systems, vehicle platooning and other interconnected systems, it is useful to develop results for models that include decentralized observer-based controllers. Consider the i’th interconnected system [3], [4], [5], [6], [9] x˙ i

= Ai xi + Bi ui + Gi hi (t, x), yi = Ci xi ,

(6)

where i = 1, 2, 3, . . . , N , xi ∈ "ni are the states, ui ∈ "mi are the inputs, yi ∈ "pi are outputs and hi (t, x) are the interconnections. It is assumed that (Ai , Bi ) is stabilizable, (Ci , Ai ) is detectable, and the uncertain term hi (t, x) is bounded by a quadratic inequality [3], [4], [5], [6], [9] hTi (t, x)hi (t, x) ≤ α2i xT HiT Hi x, (7) & % T T , αi > 0 are interconwhere x = x1 xT2 . . . xTN nection parameters and Hi are known constant matrices. For the i’th subsystem, the closed loop has the form   Ai + Bi Ki Dli Ci Cli z˙li =  0 Ai − Dli Ci −Cli  zli + 0 Bli Ci Ali   0  Gi  wi = Aˆcli zli + Gli wi , (8) 0

5602

FrB20.4

z˙N

AˆD zN + GD w(t, zN ),

=

(9)

where AˆD = diag(Aˆcl1 , Aˆcl2 , Aˆcl3 . . . , AˆclN ), GD = % &T T diag(Gl1 , . . . , GlN ) and w = w1T . . . wN satisfying ) 'N ( 1 T T zN . (10) wT (t, zN )w(t, zN ) ≤ zN 2 Hil Hil γ i=1 i

Remark 2.3: For the two coupled inverted penduli example in [3] with Bi = I2×2 in [3] replaced by Bi = % &T 0 1 , Theorem 2.1 gives / / . . −30.60 −9.23 0 0 K1 0 = K= 0 0 −30.60 −9.23 0 K2

and α1 = α2 = 0.581. The controller and observer eigenvalues has the admissibility region (amin = 0.5, ωmax = 6.3, ξ = 0.707) and (2.5, 10, 0.707), respectively. The stabilizing effect of controllers are shown in Fig. 1. As expected, the individual decoupled closed-loop subsystems, which are obtained by using K and setting hi (t, x) ≡ 0, are all stable for i = 1, 2, thereby revealing connective stability. angular pos. (pendulum 1)

The elements of Hil corresponding to xl1 , . . ., xlN are zero. In the following theorem, we provide sufficient conditions for the existence of stabilizing decentralized controllers. Theorem 2.1: If the following optimization problem is feasible N ( γi , min

1.5

angular vel. (pendulum 1)

% T &T x ˆi eTi xTli where zli = and xli ∈ "nli are the states of observer for the i’th subsystem. Defining % &T ˆTN eTN xTlN zN = xˆT1 eT1 xTl1 . . . x , the overall closed loop system can be written as

1 0.5

0

    

ΠT 2

D

,

ˆD YD + YD A ˆT Π2 A D D

ΠT 2D GD −I

... ... .. .

ΠT2D YD Π2D > 0,  T T

Π2 YD HN D l 0 0 . . . −γN I

   < 0,  

then the system in (9) is asymptotically stable for all nonlinearities satisfying the quadratic constraint in (10). Proof: Please see Appendix. Corollary 2.1: Under the conditions of Theorem 2.1, if the quadratic constraint in (7) or (10) holds globally, then the system in (9) is globally stabilized. Remark 2.1: Theorem 2.1 does not place any constraint on the closed loop eigenvalues. There is a demand of constraining the position of closed loop eigenvalues to fasten the observer dynamics, to avoid low-damped and/ or highfrequency modes and to overcome very high feedback gain Ki which the LMI formulation may produce. Therefore, after applying some congruence transformations on the pole placement objectives of [19], additional constraints are added to bound the minimum decay rate, the maximum overshoot and the frequency of oscillatory modes for the control signal and the observation error dynamics, respectively. Remark 2.2: It is interesting to note that the design algorithm in this section recognizes the matching conditions by returning an observer-based controller for any bound on the interconnection terms. Consider the system in (1) with G = B, and where hr (t, x) is replaced by g(t, x), with g(t, x) satisfying g T (t, x)g(t, x) ≤ α2 xT HrT Hr x. It is well known that when matching condition is present, there always exists a stabilizing feedback control regardless of the size of perturbations [18]. It can be easily shown that our LMI formulation recognizes it and a stabilizing observer based controller can always be computed by the LMI method. This is also true for the interconnected system in (6). The proof is omitted here for space limitations.

5 Time[sec]

10

3

−4

angular vel. (pendulum 2)

s.t

+

angular pos. (pendulum 2)

i=1



0

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2 1

0

5 Time[sec]

10

0

5 Time[sec]

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4 2 0

−2

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Fig. 1.

10

−6

Stabilizing effect of controllers

Remark 2.4: Decentralized control design has a deficiency that it cannot be applied to systems having unstable fixed modes [20]. In some cases, the block diagonal Lyapunov function may also lead to infeasibility of the LMIs. Remark 2.5: It is worthwhile to note that due to the dynamic observer, the number of1 LMI decision variables are 2/ 0N . 3(ni +1) . n li × (nli + pi ) + ni × nli + mi + pi + i=1 2 In 0N[3], [5], [6], the number of decision variables are only i=1 [ni × (ni + 1) + ni × pi + ni × mi ]. An increase in the number of decision variables compared to static observer based control design implies an increase in the off-line as well as online computational effort. III. A PPLICATION TO I NDUSTRIAL B OILER S YSTEMS In this section, we extend the nonlinear model of [17] to include the dynamics of the governor, turbine, and the generator unit. Starting from the nonlinear model in [17], some simplifications give [9]: (e22 − e12 hf )u1 + (e12 hs − e22 )u2 − e12 u3 /hm , x˙ 1 = e11 e22 − e12 e21 (e11 hf − e21 )u1 + (e21 − e11 hs )u2 + e11 u3 /hm x˙ 2 = , e11 e22 − e12 e21 e21 e32 − e11 e32 hf hc qdc x3 + u1 + x˙ 3 = − e33 e11 e22 e33 − e12 e21 e33 (e11 e32 hs − e21 e32 )u2 + (e11 e22 − e12 e21 )u3 /hm , e11 e22 e33 − e12 e21 e33 e43 hc qdc x3 ρs x4 ρs x04 e43 u3 /hm x˙ 4 = − + − . (11) e33 e44 Td e44 e44 Td e33 e44

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FrB20.4 In power generating systems, the main part of power (70%) is generated by the intermediate and low pressure part of the turbine. Therefore, we estimated the output power from the enthalpy drop in pressure interval 1 MPa to 4 MPa, which 1/8 gives ∆h+≈ 103.99x2, . Combining all, (12) reduces to 9/8 Pm = α4 Xx2 − α5 , and

Fig. 2.

Schematic of boiler, governor and turbine unit

The state variables of this system are: total water volume (x1 ), drum pressure (x2 ), steam-mass fraction in risers (x3 ), and steam volume in the drum (x4 ). The control variables are feedwater flow rate (u1 ), steam flow rate (u2 ) and fuel flow rate (u3 ). Here, e11 ,. . ., e44 are functions of the states (x1 , . . ., x4 ), the construction parameters and the properties of steam [17]. Linearization of (11) shows that the system has two poles at the origin, one associated with water dynamics and the other with pressure dynamics [17], [9]. In the following, we move the pole associated with the pressure dynamics to the left half plane. This is done by connecting the boiler drum to a governor, turbine, and generator unit (Fig. 2). We utilized some ideas from [21], [17], and started with some experiments that can be used as a basis of modeling. In the experiment, the variables such as the feedwater flow, fuel flow and the control valve position (X) are considered as inputs. The recorded outputs are drum level, drum pressure, steam flow, and the active power output (Pm ). It has been found that the drum pressure responds in more or less the same way to changes in fuel flow and control valve opening. However, the response of active power output to step change in control valve setting shows very fast change. Also, the drum pressure decreases with the increase in valve opening. But, these variations are not uniform with the corresponding change in input. Many other experiments also reveals that the system is highly interacting and nonlinear. As the drum pressure is a significant measure of the state of boiler, if the mass content of the drum is constant, and the distribution of energy stored in iron, water and steam masses do not change during transients, then the stored energy can be expressed as [21] E = ax2 + b. With dx2 this assumption, we have dE dt = a dt = Pi − Pm , where Pi is the input power and Pm is the output power of the boiler turbine unit. As the output power Pm is a function of control valve position and the pressure at turbine, we can write [21] Pm = r1 u2 ∆h + r2 ,

(12)

where ∆h is the enthalpy drop across the turbine, and r2 is added to account for losses in flow. Moreover, as the input power is a function of feedwater flow and fuel flow rate, Pi can be expressed as Pi = a1 u3 − a2 u1 . From the figure of steam flow u2 versus drum pressure x2 , we obtain u2 = 0.0904 × 10−3Xx2 − 538.9242, where x2 is in Pascal.

+ , dx2 1 dE 9/8 = = α2 u3 − α3 u1 − α1 Xx2 − α5 , (13) a dt dt where α1 = αa4 , α2 = aa1 , α3 = aa2 . Fig. 3 shows the dynamics of drum pressure, when small perturbations are done on steam valve opening and feedwater flow. Keeping 2 u1 and u3 constant, a change ∆x˙ 2 of dx dt at time t1 due 9/8 to change ∆X of X is ∆x˙ 2 (t1 ) = −α1 x2 (t1 )∆X(t1 ). From the figure, we get α1 = 2.876 × 10−6 . To compute α3 , small perturbations are done on feedwater flow (Fig. 3) and α3 = 26.75 Pa/kg is achieved. Similar arguments give α2 = 1019, and α5 = −9.16. After getting the parameters and substituting u2 (in terms of x2 and X), (11) and (13) clearly reveals that the pole associated with the pressure dynamics has moved to the left half of complex plane. A Taylor series expansion of (11) is then done to bring it in the standard form. As the steam mass fraction is always less than 100%, defining a region: Ω = {x : x1 , x4 ∈ ", |x3 | ≤ 1, x2 ≤ 7.1 MPa, |X| ≤ 0.5}, and denoting x0 as the operating point, we have 3 1.3(ρs − ρw )qdc {1+ hT1 (x)h1 (x) ≤ ρs [(1 − x3 )ρs + x3 ρw ] 452 x23 (ρs − ρw )2 x3 (ρs − ρw ) + [(1 − x3 )ρs + x3 ρw ] [(1 − x3 )ρs + x3 ρw ]2 x0 3 6 452 2ρs qdc (ρs − ρw ) x3 ρw ×∆x23 + 1 + Vr ρw [ρs − x3 ρw ]2 [ρs − x3 ρw ] x0 ×∆x23 + 1.4641∆X 2, ∀x ∈ Ω.

The physical model of governor, turbine and generator is: % & ω TE X˙ = −X − + u, y2T = δ ω Pm X , R 9/8 0.35P˙ mh = −Pmh + 3.12Xx2 , Pmh Pm PmI Pm , P˙ mL = − L + , P˙ mI = − I + 7 7 0.5 0.5 KP ω ¨ ω) + [0.3(Pmh + PmL ) + 0.4PmI ], δ(= ˙ =− TP TP where Pm = 0.3(Pmh + PmL ) + 0.4PmI is the total mechanical power output and subscripts h, I and L refers to high, intermediate and low pressure turbines, respectively. X is the control valve opening, δ is the power angle, ω is the angular speed, Tp is the time constant and KP is the power system gain. Control Strategy: After completing the modeling part, our control strategy is to decompose the plant into two subsystems: a) boiler and b) governor, turbine, generator unit. They are interconnected as in Fig. 2. The overall linearized system has two poles at the origin: one associated with water dynamics in boiler and the other with generator

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slope 800

t1 900 1000 1100 1200 Time[sec]

0.8 0.6 0.4 0.2

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900 1000 1100 1200 Time[sec]

Fig. 3.

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1000 2000 Time[sec]

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1000 2000 Time[sec]

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e2boiler

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Feedwater flow (kg/sec)

Control valve setting

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6700 6690

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e1turbine

6960

6710

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−0.5 2 e3turbine

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e3boiler

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Drum pressure(kPa)

7040

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1 1.5 Time [sec]

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Computation of parameter α1 and α3 Fig. 5.

Estimation error dynamics

dynamics. Moreover, the electrical unit has also eigenvalues at 0.9359 ± 8.0259i. The stabilizing decentralized controller is designed by solving the linear objective minimization problem of Theorem 2.1. It is given by: Kturbine =

0 2

0

5

10

0 −2 2

0

0.5

1

1.5

2

1 0 0.02

0

500

1000

0.01 0

Speed (rad/sec)

1

ˆ + z T P Gr w < 0. P > 0, z T AˆT P z + wT GTr P z + z T P Az

0

−5

Steam vol. (m3) Pressure (MPa) Valve (%)

Steam (%)

Water vol. (m3) Power (MW)

Angle (rad)

eigenvalues, and the bounds on nonlinear terms. It is useful to know that the proposed LMI formulation maximizes the interconnection bounds and the control law results in a connectively stable system, thus overcoming the barrier of [ −0.2671 −0.4325 −1.0595 −1.7186 −1.0506 −4.4705 ] , instability caused by sudden structural perturbations. The 3 5 −119.48 0.000253 −7391.596 −152.91 Kboiler = . developed theoretical framework is then applied to a power −3.136 −0.0121 −194.039 −4.0141 unit. Simulation results show that the stabilizing effect of controllers are good both during normal and perturbed conThe stabilizing effect of controllers are shown in Fig. 4. It ditions, which make it practically implementable. can be seen that the dynamics of electricity generating unit is much faster than the drum-boiler dynamics. The term ‘0’ on V. A PPENDIX 0 the y-axis represents steady state values (X 0 = 0.5, Pm = Proof of Theorem 2.1: Here, we first develop a control design 48MW etc.). Fig. 5 shows the dynamics of estimation error, algorithm for the autonomous system in (1), which is then where (e1 )boiler , (e2 )boiler and (e3 )boiler represents x1 − xˆ1 , generalized to multiple subsystems in (6). Let us consider a x2 − x ˆ2 and x4 − x ˆ4 , respectively.. The others are for the Lyapunov function v = z T P z [13], where P is a symmetric turbine-generator unit (δ, ω and Pm ). positive definite matrix (P > 0). The sufficient conditions for asymptotic stability of the closed loop system in (4) are 2 5

2

0

5

According to the S-procedure [14], when (5) is satisfied, the above condition is equivalent to the existence of matrix P and a number τ > 0 such that P > 0 and 3 T 5 Aˆ P + P Aˆ + τ α2 H T H P Gr < 0. (14) GTr P −τ I

10

0

−2 1

0

0.5

1

1.5

2

0

−1 0

0

5

10

0

500 Time [sec]

1000

This is equivalent to the existence of matrix Y > 0, and   ˆ + Y AˆT Gr Y H T AY  (15) GTr −I 0  < 0, HY 0 −γI

−1

0

500 Time [sec]

Fig. 4.

1000

−2

Stabilizing effect of the controller

IV. C ONCLUSIONS In this paper, we proposed a new algorithm for decentralized observer based control design, which guarantees robust stabilization of the overall interconnected system. The approach alleviates the necessity of having invertible input matrix Bi [3], choice of parameters by trial and error [5] or any two step approach [6]. Our design has the flexibility of accommodating various design constraints involving matching conditions, separate regions of controller and observer

where Y = τ P −1 and γ = α12 . This LMI cannot be used to find the observer-based controller because it is not affine on observer parameters Al , Bl , Cl and Dl , which are constant nl × nl , nl × p, n × nl , and n × p matrices, respectively. Hence, a variable transformation is necessary. Partitioning     R1 0 0 R2 0 0 Xr Mr  , Y −1 =  0 Yr Nr  , Y = 0 0 MrT Vr 0 NrT Ur

where R1 , R2 , Xr and Yr are n × n and symmetric, Mr and Nr are n × nr and Y > 0 implies R1 > 0, Xr >

5605

FrB20.4 −1 0, and % YrT > 0.T From Y&T Y %= I, it can&Tbe inferred that −1 R1 Xr Mr Y = I I 0 , which leads to Y −1 Π1 = Π2 , where,     R1 0 0 I 0 0 Xr I  , Π2 !  0 I Yr  . (16) Π1 !  0 0 MrT 0 0 0 NrT

Pre-and post-multiplying Y > 0 by ΠT2 and Π2 , respectively and (15) by diag (ΠT2 , I, I) and diag (Π2 , I, I), respectively, we have ΠT2 Y Π2 > 0 and  T  ˆ Π2 + ΠT Y AˆT Π2 ΠT Gr ΠT Y H T Π2 AY 2 2 2   < 0. (17) GTr Π2 −I 0 HY Π2 0 −γI Some linear algebra shows that   R1 0 0 ΠT2 Y Π2 =  0 Xr I  and 0 I Yr 

ˆ Π2 ΠT2 AY

=

AR1 + BL1  0 0

ˆr C ˆr AXr − C ˆ Ar



ˆrC D ˆ r C , A−D ˆr C Yr A + B

Aˆr ! YrT (A − Dl C) Xr + Nr Bl CXr − YrT Cl MrT + ˆr ! −Yr Dl + Nr Bl . Nr Al MrT , B (18) F11 ˆ Π2 + ΠT Y AˆT Π2 =  Cˆ T Also, ΠT2 AY 2 1 ˆT CT D 1   0 ΠT2 Gr =  G  , and ΠT2 Y H T Yr G

Pre-and post-multiplying YD by ΠT2D and Π2D , respectively, and the above constraint by diag (ΠT2D , I, I) and diag (Π2D , I, I), respectively, the optimization problem become affine in controller and observer parameters. " R EFERENCES

ˆ r ! Dl , Cˆr ! Dl CXr + Cl MrT , where L1 ! KR1 , D



diag(Π21 , . . . , Π2N ). Similar to (15), it is easy to verify that the sufficient conditions of the closed loop system in (9) to be asymptotically stable under the constraint in (10) are YD > 0, and   T AˆD YD + YD AˆTD GD YD H1Tl . . . YD HN l   −I 0 ... 0     −γ1 I . . . 0   < 0.   .. ..   . . −γN I

ˆ 1C  D F23  , F33  R1 HrT =  Xr HrT  , HrT Cˆ1 F22 T F23 

where F11 ! AR1 + BL1 + R1 AT + LT1 B T , F22 ! ˆ r C and AXr + Xr AT − Cˆr − CˆrT , F23 ! A + AˆTr − D ˆr C + AT Yr + C T B ˆrT . This makes the F33 ! Yr A + B LMIs in (17) affine in controller and observer parameters. A linear objective minimization (min γ) subject to the convex constraints in (17) can be easily applied to compute them. Since Xr and Yr are symmetric matrices, Nr and Mr can be chosen square and non-singular such that Nr MrT =% I − Y&r Xr . Using singular value decomposition, we have ΣΛΩT = svd (I − Yr Xr ) . This gives, Nr MrT = 1 1 ΣΛΩT , Nr = ΣΛ 2 , Mr = ΩΛ 2 . Hence, from (18), the parameters K, Al , Bl , Cl and Dl can be easily calculated. This method does not require that the input matrix B matrix be invertible [3], or choice of any parameters by trial and error [5]. Hence, the critical restriction is removed. The trick lies in congruence transformations using (16) and defining new variables as in (18). One can easily verify that when Al = Bl = Cl = 0 (static observer gain), then ˆr = −Yr Dl , Cˆr = Dl CXr and Aˆr = YrT (A − Dl C) Xr , B ˆ r = Dl . Hence, the parameters become non affine. D Now, the decentralized control design for the system in (6) is quite straightforward with block diagonal Lyapunov function YD = diag(Y1 , . . . , YN ) and Π2D =

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