Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844 Vol. V (2010), No. 3, pp. 385-397 Robust Control of Particle Size Distribution in Aerosol Processes Z. Xiang Zhengrong Xiang School of Automation Nanjing University of Science and Technology Nanjing, 210094, People’s Republic of China E-mail: xiangzr@mail.njust.edu.cn Abstract: This paper deals with a comprehensive study on robust control of particle size distribution of fractal agglomerate in aerosol processes with simultaneous chemical reaction, nucleation, condensation and coagulation. Firstly, a general aerosol process is described by population balance and mass and energy balances, which describes the evolution of particle size distribution, continuous phase species and temperature of the aerosol system, respectively. A lognormal moment ap- proximations of the population balance model is then presented. Then, the robust state feedback controller is designed for the aerosol process with some unknown uncertainties, the proposed controller is composed of an nominal control term and a robust control term so that it only ensures the stability of the closed-loop system, but also attenuates the effect of the unknown uncertainties on the system. A high-gain observer is adopted to estimate state variables required in the on-line implementation of the state feedback. Finally, the proposed robust controller is applied to an aerosol process for achieving an aerosol size distribution with desired geometric average particle diameter, the simulation results show the robustness properties of the controller with respect to parametric model uncertainty and unmodeled dynamics. Keywords: particle size distribution, aerosol process, population balance, nonlinear systems, robust control 1 Introduction Aerosol process is of interest in many problems of ecology, atmospheric physics, and mechanics of multiphase system[1-3]. The evolution of an aerosol size distribution is modeled by population balance equations, which give rise to nonlinear partial integro-differential equation systems. The nonlinearities usually arise from complex reaction, nucleation, condensation and coagulation rates and their nonlin- ear dependence on temperature. A variety of solution techniques have been developed to address the complexity at various levels. In most cases, one needs to resort to numerical solutions. One of the stan- dard numerical techniques is to discretize the population balance equation using finite difference/element methods (see[4-8], and the references cited therein), but these methods suffer from extremely large com- putational requirements which cannot be accommodated by conventional computers. Sectional models offer a computationally less demanding solution by approximating the continuous size distribution by a finite number of sections within which the particle size distribution (PSD) function is assumed to be constant [9]. On the other hands, under appropriate simplifying assumptions, analytical solutions have been developed to solve the population balance equation [10-13]. Recently, the issue of population balance mode-based feedback control of PSD has received consider- able attention [14-16]. Previous work in this area include stability analysis and the application of the con- ventional control schemes to crystallizers and emulsion polymerization processes [17]. Unfortunately, conventional control schemes perform poorly in the face of severe process nonlinearities, and may even Copyright c© 2006-2010 by CCC Publications 386 Z. Xiang lead to destabilization of the closed-loop system. These limitations of conventional control schemes have motivated research efforts towards synthesizing nonlinear model-based feedback controllers on spatially- homogeneous aerosol processes with application to size distribution control in continuous crystallizers [18,19], and titania aerosol reactor [20,21]. However, most real aerosol systems have uncertainties include unknown or partially known time- varying process parameters, exogenous disturbance, and unmodeled dynamics. It is well known that the presence of uncertain variables and unmodeled dynamics, if not taken into account in the controller design, may be lead to severe deterioration of the nominal closed-loop performance or even to closed- loop instability. Therefore, it is very important thing to study robust control of nonlinear aerosol systems with uncertainty. This paper focuses on robust control of particle process described by uncertain population balances. The objective is to develop a general method for the synthesis of practically implementable robust nonlin- ear controller that explicitly handle time-varying uncertain variables (such as unknown process parame- ters and disturbances) and unmodeled dynamics. The robust nonlinear controller enforces stability in the closed-loop system and attenuation of the effect of uncertain variables on the system, and achieve PSD with desired characteristics. To present robust output feedback control techniques that are applicable to a broad range of aerosol systems, we choose to focus on an aerosol process that involves simultaneous chemical reaction, nucleation, condensation and coagulation rather than a specific aerosol. The remainder of this paper is organized as follows: In Section 2, a general aerosol process is de- scribed by population balance and mass and energy balances, which describes the evolution of particle size distribution, continuous phase species and temperature of the aerosol process, respectively. A log- normal moment approximations of the population balance model is presented. In Section 3, the robust state feedback controller is designed for the aerosol system with some unknown uncertainties including unknown or partially known time-varying process parameters, exogenous disturbance, and unmodeled dynamics, the proposed controller is composed of a robust control term and a nominal control term so that it only ensures the stability of the closed-loop system, but also attenuates the effect of the unknown uncertainties on the system. A high-gain observer is adopted to estimate state variables required in the on-line implementation of the state feedback. In Section 4, the proposed robust controller is applied to a general aerosol process for achieving an aerosol size distribution with desired geometric average particle diameter, the simulation results show the robustness properties of the controller with respect to parametric model uncertainty and unmodeled dynamics. Finally, conclusions are given in Section 5. 2 Mathematical model of aerosol process 2.1 Spatially-homogeneous aerosol process Consider a general aerosol process which described by the following nonlinear partial integro-differential equation [22]: ∂ n(v,t) ∂ t + ∂ (G(x̄, v)n) ∂ v − I(v∗)δ (v − v∗) =   ∫ v  β (x̄, v − v̄, v̄)n(v − v̄,t)n(v̄,t)dv̄ − n(v,t) ∫ ∞  β (v, v̄)n(v̄,t)dv̄ (1) where the term n(v,t) represents the PSD function at time t , v is the particle volume, I(v∗) is the nucle- ation rate, G(x̄, v) and β (x̄, v − v̄, v̄) are the diffusional condensation growth function and the Brownian coagulation kernel of agglomerates, δ (·) is the standard Dirac function. Robust Control of Particle Size Distribution in Aerosol Processes 387 A mathematical model which predicts the time evolution of the concentrations of species and tem- perature of the gas phase has the following form [21]: dx̄ ∂ t = f (x̄) + g(x̄)u(t) + Ā ∫ ∞  a(η, v, x̄)dv (2) where x̄(t) is an n-dimensional vector of state variables that depend on time, Ā is constant matrix, f (x̄), g(x̄), a(η, v, x̄) are nonlinear vector functions and u(t) is the time-varying manipulated input (e.g. wall temperature). The term Ā ∫∞  a(η, v, x̄)dv accounts for mass and heat transfer from the continuous phase to all the particles in the population. The diffusional condensation growth function G(x̄, v) and the Brownian coagulation kernel of ag- glomerates β (x̄, v − v̄, v̄), for the free molecule size regimes, are represented by GF M(x̄, v) = Bv /(S − ), B = (π)/vns ( kBT π m )/ , βF M (x̄, v, v̄) = B ( v/D f + v̄/D f ) √  v +  v̄ , B = (/π) /D f −/ (kBT /ρ) / r−/D f (3) and for the near-continuum and the continuum regimes Gc(x̄, v) = Bv /(S − ), B = (π )/C f vns,C f = λ  ( kBT π m )/ βc (x̄, v, v̄) = B ( v/D f + v̄/D f ) [ C(v) v/D f + C(v̄) v̄/D f ] , B = kBT µ (4) In the above equations, S is the saturation ratio, T is the absolute temperature, D f is the fractal (Haus- dorff) dimension, C f is the condensable vapor diffusivity, λ is the mean free path of the gas ( λ = vπ Mw/2kBT Nav, where v and Mw are the kinematic viscosity and molecular weight of the fluid, respec- tively, and Nav is the Avogadro’s number), µ is the viscosity of the fluid, ns is the monomer concentra- tion at saturation (ns = Ps/kBT , where Ps is the saturation pressure), m is the monomer mass, v is the monomer volume, ρ is the particle density, r is the particle radius, C(v) =  + Bλ /r is the Cunningham slip correction factor, and B = .. The nucleation rate I(v∗) is assumed to follow the classical Becker-Doring theory and is given by the following expression [23]: I = ns ( kBT π m )/ S (  π )/ Σ / exp(−k∗ ln S  ) (5) where s is the monomer surface area and k∗ is the number of monomers in the critical size nucleus which is given by: k∗ = π  ( Σ ln S ) (6) where Σ = γ v/ /kBT and γ is the surface tension. Remark 1. It is found experimentally that in many cases of practical interest, the total number of primary particles Np in an agglomerate is related to characteristic radius R through a power law expression Np ∼ RD f , where the exponent D f is called the fractal dimension. This is usually true in a statistical sense after averaging over many agglomerates with the same Np. The value of D f depends on the details of the agglomerate formation process. For compact agglomerates we have D f → , while for chain-like agglomerates we have D f →  . 388 Z. Xiang Remark 2. It is noticed that the collision kernel βF M in (3) is in a form that appears to be rather difficult to expand into a series with a manageable number of terms. The following approximation can be made [11]; (  v +  v̄ )/ = b (  v/ +  v̄/ ) The variables for b depend on the initial geometric standard deviation σ and on the fractal dimension D f . Remark 3. The manipulated input u(t) enters system (1)-(2) through (2), this assumption is usually satisfied in most practical applications where the wall temperature is chosen as the manipulated input. 2.2 Moment model In this subsection, we present moment model of size distribution of fractal agglomerates in aerosol processes with simultaneous chemical reaction, nucleation, condensation and coagulation. Many ex- perimental results and numerical calculation indicate that aerosol PSDs can be adequately described by unimodal lognormal functions. This result makes it possible to develop a moment model for the aerosol process in terms of the three leading moments of the size distribution. The log-normal size distribution function is written as: n(v,t) =  v N(t)√ π ln σ (t) exp [ − ln{v/vg(t)}  ln σ (t) ] where N(t) is the total number concentration of particles, σ (t) is the geometric standard deviation, and vg(t) is the geometric number mean particle volume. The kth moment of the particle size distribution is written as: Mk(t) = ∫ ∞  vkn(v,t)dv where k is an arbitrary real number. According to the properties of a log-normal function, any moment can be written in terms of M, vg and σ as follows Mk = Mv k− g exp {   (k − ) ln σ } (7) Obviously, the three leading moments are sufficient to generate the lognormal PSD. Using the same derivation procedure as the one in [21], we can easily obtain the following equations, which describe the time evolution of the three leading moments (i.e. the zeroth, first and second moments) of the size distribution for the free molecule size. dM dt = I − bB(M/D f −/M + M/D f M/D f −/ + M/D f M−/) (8) dM dt = Iv∗ + B(s − )M/D f −/ (9) dM dt = Iv∗ + B(s − )M+/D f +bB(M/D f +/M + M+/D f M/D f +/ + M+/D f M/) (10) Similar to the case of the free molecule regime, the dynamics of the zeroth and second moments of the aerosol size distribution in the continuum size regime is described by the following equations: dM dt = I − B[M   + M/D f M−/D f + Bλ (   π )/ (MM−/D f + M/D f M−+/D f )] (11) dM dt = Iv∗ + B(s − )M/D f (12) Robust Control of Particle Size Distribution in Aerosol Processes 389 dM dt = Iv∗ + B(s − )M+/D f +B[M   + M+/D f M−/D f + +Bλ (   π )/ (MM−/D f + M/D f M+/D f )] (13) Table 1 Dimensionless variables N = M/ns, Aerosol concentration V = M/nsvd, Aerosol volume V = M/nsv, Second aerosol moment τ = (π m/kBT )//ns, Characteristic time for particle growth K = (kBT /µ)nsτ , Coagulation coefficient Kn = λ /r, Monomer Knudsen number I ′ = I/(ns/τ), Nucleation rate R′r = Rr/(ns/τ), Reaction rate group v′g = vg/v r ′g = rg/r θ = t/τ The moment equations for the free molecule size and continuum size regimes may be combined to describe the aerosol dynamics over the entire particle size spectrum by using the harmonic average of the dimensionless coagulation rates in the free molecule size and continuum size regimes [23]. Using the dimensionless variables listed in Table 1, we have to the following equations: Zeroth moment dN dθ = I ′ − ξ N (14) where ξ = ξF M ξc ξF M + ξc , ξF M = r ′/ g b [ exp (   ln σ ) +  exp (   ln σ ) + exp (   ln σ )] ξc = K [  + exp (ln σ ) + B ( Kn r ′g ) exp (   ln σ ) ( + exp ( ln σ ) ] First moment dV dθ = I ′k∗ + η(S − )N (15) where η = ηF M ηc ηF M + ηc , ηF M = v ′/ g exp ( ln  σ ) ηc = Kn  v ′/ g exp (   ln σ ) Second moment dV dθ = I ′k∗ + ε(S − )V + ζ V  (16) where ε = εF M εc εF M + εc , ζ = ζF M ζc ζF M + ζc εF M = v ′/ g b exp ( ln  σ ) , εc = Kn  v′/g exp (   ln σ ) ζF M = r ′/ g b exp (   ln σ ) [ exp (   ln σ ) +  exp (   ln σ ) + exp (   ln σ )] ζc = K [  + exp (ln σ ) + B ( Kn r ′g ) exp ( −   ln σ ) ( + exp (− ln σ ) ] 390 Z. Xiang The rate of change of S can be obtained from a monomer balance and is given by: dS dθ = R′rNav − I ′k∗ − η(S − )N (17) 2.3 Mathematical model of aerosol process We consider a typical aerosol process in a cylindrical volume with diameter DT . Under the assump- tion of lognormal aerosol size distribution, the dimensionless model that describes the evolution of the first three moments of the distribution, along with the saturation ratio, reactant concentrations and fluid temperature, takes the following form: dN dθ = I ′ − ξ N dV dθ = I ′k∗ + η(S − )N dV dθ = I ′k∗ + ε(S − )V + ζ V  dS dθ = CC̄C̄ − I ′k∗ − η(S − )N dC̄ dθ = −AC̄C̄ dC̄ dθ = −AC̄C̄ dT̄ dθ = BC̄C̄T̄ + ET̄ (T̄w − T̄ ) (18) where C̄ and C̄ are the dimensionless reactant concentrations, T̄ , T̄w are the dimensionless fluid tem- perature and the dimensionless temperature of the heat transferring medium at the system boundary, respectively. A, A, B,C, E are dimensionless quantities, their explicit form are given in Table 2. Table 2 Dimensionless variables for the model A = τ kPy/RT A = τ kPy/RT B = τ kP∆ Hτ yy/RT Cp C = Navτ kyy(P/RT)/ns E = U RTτ/DT CpP C̄i = yi/yiT̄ T̄ = T /T T̄w = Tw/T We formulate the control problem as one of tracking the geometric average particle diameter of the aerosol system, by manipulating the wall temperature, i.e. y(t) = dpg(t), u(t) = T̄w(t) − T̄ws where T̄ws = Tws/T = . We write the model (18) in the following form: dx̃ dt = f̃ (x̃) + g̃(x̃)u(t), y = h̃(x̃) (19) where the explicit form of x̃, f̃ (x), g̃(x̃) can be obtained by comparing (18) and (19). Robust Control of Particle Size Distribution in Aerosol Processes 391 Remark 4. In addition to being highly nonlinear, the real aerosol processes have uncertainties including unknown or partially known time-varying process parameters, exogenous disturbance, and unmodeled dynamics. Therefore, the real aerosol systems can be described by the following systems. dx̃ dt = f̃ (x̃) + ∆ f̃ (x̃) + (g̃(x̃) + ∆ g̃(x̃))u(t), y = h̃(x̃) (20) where ∆ f̃ (x̃) and ∆ g̃(x̃) represent uncertainties caused by unknown or partially known time-varying process parameters, exogenous disturbance, and unmodeled dynamics. 3 Robust controller design In this section, we will begin with the design of the robust controller. Our objective is to design a robust output feedback controller which guarantees boundedness of all variables for the closed-loop system and tracking of a given reference signal yd . By differentiating the output y(t) with respect to t , we obtain the following nonlinear system repre- sented by the differential equation y(r) = f (y, y(),··· , y(r−)) + ∆ f (y, y(),··· , y(r−))+ (g(y, y(),··· , y(r−)) + ∆ g(y, y(),··· , y(r−)))u (21) where r is the relative order (The definition can be found in [24], and omitted here for brevity) of the output y(t) with respect to the manipulated input u(t) . By taking x = y, x = y() up to xr = y(r−), we can represent the augmented system by the state space model ẋ = Ax + B(v + D(x)) y = x (22) where x = (x, x,··· , xr)T , v = f (x) + g(x)u, δ(x) = ∆ g(x)/g(x), D(x) = ∆ f (x) + δ(x)(v − f (x)), (A, B) is controllable canonical pairs of the form A =      ···     ···  ··· ··· ··· . . . ...    ···     ···    ∈ Rr×r, B =     ...     ∈ Rr× Assuming the states is available for feedback, taking e = y − yr = x − yr e = ẏ − ẏr = x − ẏr ... er = y(r−) − y (r−) r = xr − y (r−) r and e = (e, e,··· , er)T , we rewrite (22) as ė = Ae + B(v + D(e + Yd ) − y (r) d ) (23) where Yd = (yd , ẏd ,··· , y(r−)d )T . 392 Z. Xiang Obviously, if D(·) =  (i.e. ∆ f (x) = ∆ g(x) =  ), we can use the following nominal state feedback control to achieve our objective. u =  g(x) (−Ke + y(r)d − f (x)) (24) where matrix K is chosen such that the matrix Am = A − BK is Hurwitz, i.e., all its eigenvalues are in the left open plane. However, due to the presence of uncertainties ∆ f (x) and ∆ g(x) , a nominal state feedback control alone cannot ensure the stability of the closed-loop system, it is necessary to add a robust compensator to deal with it. u = −  g(x)( − ∆(x)) BT Peµ (x) ‖BT Pe‖µ(x) + ε exp(−εt) (25) where ε, ε > , P = PT >  is a solution of the following Lyapunov equation P(A − BK) + (A − BK)T P = −Q, ∀QT = Q >  µ(x) = ∆(x) (∣∣∣y(r)d − Ke ∣∣∣ + | f (x)| ) + |∆ f (x)| , ∆(x) = sup x∈Rr |δ(x)| The robust state feedback controller we proposed is as follows u = u + u (26) Theorem 5. Suppose that ∆(x) = sup x∈Rr |δ(x)| < , consider system (22) under the robust state feedback controller (26), the closed-loop system have the following properties: (1) The state of the closed-loop system is bounded. (2) The output of the closed-loop system satisfy lim t→∞ |y − yd | = . Proof: Let ν = −  ( − ∆(x)) BT Peµ(x) ‖BT Pe‖µ(x) + ε exp(−εt) The closed-loop system under the robust state feedback controller (26) is as follows ė = (A − BK)e + B(v + D(x)) (27) Consider the Lyapunov function candidate V (e) = eT (t)Pe(t) The derivative of V (e) along the trajectories of the system is given by V̇ (e) = eT P(A − BK)e + eT PB(v + D(x)) (28) Obviously, we have |D(x)| ≤ |∆ f (x)| + ∆(x)(|v| + | f (x)|) ≤ ∆(x) |v| + |µ(x)| Therefore eT PB(v + D(x)) ≤ − ( − ∆(x)) ∥∥BT Pe ∥∥· |v| +  ∥∥BT Pe ∥∥ µ(x) Robust Control of Particle Size Distribution in Aerosol Processes 393 ≤ B T Peµ(x)ε exp(−εt) ‖BT Pe‖µ(x) + ε exp(−εt) ≤ ε exp(−εt) (29) Substituting (29) into (28), we have V̇ (e) ≤ −λmin(Q)‖e‖ + ε exp(−εt) (30) where λmin(Q) denotes the minimum eigenvalue of Q . Using the lemma in [25], we have lim t→∞ e =  This completes the proof. 2 The on-line implementation of the controller (26) requires that the values of the state variables x are known. Unfortunately, x may be not be known in many practical applications. One way to address this problem is to use state observer to estimate x. Here we use the high-gain state observer as ˙̂ei = êi+ + αi ε (e − ê),  ≤ i ≤ r −  ˙̂en = αn ε (e − ê) (31) where ê = x̂ − Yd , x̂ denote the estimate state of the state x, ε is a small positive parameter, the constants αi >  are chosen such that the roots of sr + αsr− + ···+ αr−s + αr =  have negative real parts. The state feedback controller (26) and the state observer (31) can be combined to yield the following nonlinear robust output feedback controller: ˙̂ei = êi+ + αi ε (e − ê),  ≤ i ≤ r −  ˙̂en = αn ε (e − ê) u =  g(ê + Yd ) ( y(r)d − Kê − f (ê + Yd ) −  ( − ∆(ê + Yd )) BT Pêµ (ê + Yd ) ‖BT Pê‖µ(ê + Yd ) + ε exp(−εt) ) (32) Theorem 6. Suppose that ∆(x) = sup x∈Rr |δ(x)| < , consider system (22) under the robust state feedback controller (32), the closed-loop system have the following properties: (1) The state of the closed-loop system is bounded. (2) The output of the closed-loop system satisfy lim t→∞ |y − yd | =  . We can prove Theorem 2 by following the proof line of Theorem 1, the detailed proof is omitted here. 4 Simulation study The performance of the nonlinear robust output feedback controller (28) was tested through simula- tions. The values of the process parameters are shown in Table 3. Table 3 Model parameters for the simulation study 394 Z. Xiang DT = .m, Process diameter P = atm, Process pressure T = K, Process pressure Y = Y = ppm, Initial mole fraction of reactants U = .Jm−s−K−, Overall coefficient of heat transfer ∆ Hr = .KJml−s−, Heat of reaction Cp = .Jmol−K−, Heat capacity of process fluid Mw = .×−kgmol−, Molecular weight of process fluid k = .mmol−s−, Reaction constant µ = .×−kgm−s−, Viscosity of process fluid γ = .Nm−, Surface tension v = .×−m, Molecular weight of process fluid R = .Jmol−K−, Universal gas constant Nav = .mol−, Avogadro’ constant kB = .×−JK−, Boltzmann’s constant Several simulations were performed to evaluate the disturbance attenuation and set-point tracking capabilities of the robust nonlinear controller, as well as its robustness with respect to uncertainties in model parameters and unmodeled dynamics. The objective of these simulations is to show that the use of proposed robust control allows producing an aerosol product with a desired geometric average particle diameter (dpg = .µ m) within a given type of batch reactor even in the presence of uncertainty in the process model parameters. Figure 1 presents the closed-loop trajectory (dashed line) and closed-loop trajectory (solid line) for dpg under robust nonlinear controller and nonlinear controller, respectively, and the corresponding manipulated input trajectory in the case of 10% error in the value of the rate constant k. It is clear that the robust nonlinear control regulates the output dpg to the desired point value (i.e. dpg = .µ m), and attenuate the effect of time- varying uncertainty on the process. However, nonlinear control cannot guarantee the output dpg to the desired point values in the presence of uncertainty. There is an error between steady-state value and the desired point one. Figure 2 presents the closed-loop trajectory (dashed line) and closed-loop profile (solid line) for dpg under robust nonlinear controller and nonlinear controller, and the corresponding manipulated input trajectory in the case of a 8% decrease in the parameters. Again, the robust nonlinear controller allows achieving an aerosol product with the desired dpg. Also we simulated the robustness properties of the proposed controller with respect to parametric model uncertainty and unmodeled dynamics in the presence of a set-point change, the robust nonlinear controller was found to have very good robustness properties, keeping the output on the set-point. Remark 7. It is worth to point out that the proposed approach for the design of robust nonlinear con- trollers is applicable to most aerosol systems for which the hypothesis of unimodal lognormal aerosol size distribution for long times is valid. 5 Conclusions In this paper, we present a comprehensive study on robust control of particle size distribution of fractal agglomerate in aerosol processes with simultaneous chemical reaction, nucleation, condensation and coagulation. Initially, a general aerosol process is presented by population balance and mass and en- ergy balances, which describes the evolution of particle size distribution, continuous phase species and temperature of the aerosol system, respectively. A lognormal moment approximations of the population balance model is then presented. Then, the robust state feedback controller is designed for the aerosol system with some unknown uncertainties, the proposed controller is composed of a robust control term and an nominal control term so that it only ensures the stability of the closed-loop system, but also atten- Robust Control of Particle Size Distribution in Aerosol Processes 395 Figure 1: (a) dpg using the robust nonlinear controller and nonlinear controller, (b) The corresponding manipulated input trajectory Figure 2: (a) dpg using the robust nonlinear controller and nonlinear controller, (b) The corresponding manipulated input trajectory 396 Z. Xiang uates the effect of the unknown uncertainties on the system. A high-gain observer is adopted to estimate state variables required in the on-line implementation of the state feedback. 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Yaz, Stabilizing compensator design for uncertain nonlinear systems, Systems and Control Let- ter, 21, pp.11-17, 1993 Zhengrong Xiang was born in China, in 1969. He obtained his Ph.D. degree in Control Theory and Control Engineering from the Nanjing University of Science and Technology, China, in 1998. Since 2001, he has been an associate professor in the School of Automation. He is an IEEE member. His main research interests include nonlinear control, robust control, intelligent control, and switched systems. He has published significantly on the subjects with over 80 technical papers in journals and conferences. E-mail: xiangzr@mail.njust.edu.cn.