H-infinity control with D-stability constraint for uncertain repetitive processes

Transcription

1 H-infinity control with D-stability constraint for uncertain repetitive processes Nizar OUJENI, Chaoui MNASRI, Moncef GASMI Computer Laboratory for Industrial Systems (LISI National Institute of Applied Sciences and echnology (INSA, Carthage University unisia Abstract his paper is interested in the problem of H- infinity control with D-stability constraint for uncertain continuous-time repetitive systems with eternal disturbances. he main objective is the design of a control law, such that the system closed-loop poles are placed within a particular region of the comple plane for all admissible uncertainties. All of the obtained conditions are formulated in the form of linear matri inequalities and solutions gives the agreed controller gains. Finally, a numerical eample is given to illustrate the effectiveness of the proposed approach. Keywords Repetitive control, uncertain linear systems, robust control, linear matri inequality, H-infinity and D-stability. I. INRODUCION In engineering practice, repetitive processes are very common and are usually encountered in many industrial applications such as power supply systems [-], robotic manipulators [3], CD tracing [4], computer dis drives [5-6], etc. In those applications, the control systems are usually desired tracing or rejecting periodic eogenous signals with high control precision. he repetitive control was first presented by Inoue et al. and applied to the control of a contouring servo system and a power supply for a proton synchrotron [7-8]. After that, it has been applied to many problems. he repetitive control affords a successfully practicable solution and that is a control scheme applied to systems that must cancel error, trac periodic reference signals or reject periodic disturbances. Referring to offered Wu et al. [9-3], some design methods of repetitive control system for a class of linear system based on two-dimensional continuous/discrete hybrid model are presented. he problem for the design repetitive controller is converted in a problem for a continuous-discrete twodimensional system. After that, this problem is solved by combing two-dimensional Lyapunov theory with linear matri inequalities approach. In practice, the influence of eternal disturbances and uncertainties in the plant must be strictly considered when the repetitive controller is applied to real systems. In many cases, those parameters cause instability in the control system. he stability problem with the uncertainties is named robust stability problem. Yamada and al. [4-8] were proposed some design methods for repetitive control systems with considering disturbances and uncertainties. In robust control system, stability of the closed-loop system maes the minimum specification. Sometimes, owing to bad transient responses in many applications or real physical systems, the system dynamic features do not mae the desired goals such as transient oscillations, the rise time, the settling time, etc. A satisfying performances can be achieved by placing the closed-loop pole to in an appropriate region of the comple plane. Enforcing all poles of a system in a specified region is named D-stability problem. he main contribution of this paper is to study the problem of H-infinity control with D-stability constraint for uncertain continuous-time repetitive systems with eternal disturbances for all admissible uncertainties. In the first part, we will prove an equivalence between a two-dimensional control system and a repetitive control scheme such that study of convergence and stability properties. In the second part, all of the obtained conditions are formulated in the form of linear matri inequalities and solutions gives the agreed controller gains. Finally, an eample shows the efficiency of the proposed approach will be presented. II. PROBLEM FORMULAION AND PRELIMINARIES Consider the uncertain linear system defined by the following state-space equation ɺ ( t = ( A + A ( t + ( Bu + Bu u( t + ( Bd + Bd d( t y( t = C ( t + Du u( t ( t is the state vector, u( t is the input control, y( t is the output of the system and d ( t is an eternal disturbance. A, B u, B d, C and D u are real matrices. A, Bu and Bd denote real matri functions representing norm-bounded time varying parametric uncertainties in the system model. We consider the following assumptions: (i he pair ( A, Bu is stabilizable (ii d( t is an eternal disturbance signal with finite energy in the space L [, + (

2 (iii Uncertainties under consideration have the following form A = A + A = A + HF( t EA B = B + B = B + HF( t E B = B + B = B + HF( t E F ( t. F( t < I u u u u Bu d d d d Bd I is the identity matri of appropriate dimensions. F( t is unnown real time varying matri contain uncertain parameters and H, E A, E Bu and E Bd are nown constant real matrices of appropriate dimensions denote how the uncertain parameters F( t affect the system (. he output error is defined by e( t = r( t y( t r( t = r( t + is the periodic reference and is the fundamental period. he robust repetitive control law proposed of the system is u( t = G ( t + G Φ ( t (3 rob rep e( t, t < Φ ( t = Φ ( t + e( t, t Φ( t defines the output signal of the repetitive controller and pair ( rob, rep ( (4 G G creates gain matrices to be determined. he first describes control action and the second describes the learning action. Consider now the variables N which used to describe τ, is a domain to depict learning between periods, [ [ control inside a period and ψ ( t which is described in the time domain by ψ ( t = ψ ( + τ : = ψ ψ ( t = ψ ( t ψ ( t : = ψ hen, according to ( - (5, we have ɺ = A + B u + B d (6 u d e = e - C - D u (7 u Equations (6 and (7 creates a two-dimensional (D continuous-discrete hybrid model of the repetitive control system. Net, the D control law can be written u = G + G e (8 G I + G D G G C G = ( I + Grep Du Grep =( rep u ( rob rep (5 (9 herefore, it is easy to conclude that the design of a D control law (8 is equally equivalent to design of a control law (3. hus, the control system (3 is stable if a D stabilizing control law (8 is designed for the D system (9 and the matrices gains are given by Grob = G + G ( I DuG ( DuG + C Grep = G ( I DuG ( Consequently, it is easy to adjust independently the robust control and learning actions using G and G respectively because the control action depends on both gains G and G, while the learning action depends only on G. However, it is very difficult to do that using G rob and G rep. In order to achieve the main results, some necessaries preliminaries will be introducing. Lemma. (Shur complement [9]: For any symmetric Θ Θ matri, Θ, of the form Θ =. If Θ is invertible Θ Θ then the following property hold: Θ < if Θ < and Θ - Θ Θ Θ < ( Lemma. []: Given matrices appropriate dimensions, then K = K, J, F and E of K + JFE + ( JFE < ( for all F satisfying ε > such that K ε JJ ε E E F F I, if and only if there eists some + + < (3 Definition. []: An LMI region is defined by a subset of the comple plane given by D z { z : z z } = C ϒ + Λ + Λ < (4 ϒ = ϒ and Λ are two real matrices. In this paper, LMI region chosen is the intersection of three regions given by a Re( z + b Im( z < D : conical sector : a h = arctan b D : dis of radius r centered at (q, D 3: α stability : Re( z < α

3 heorem. []: Let Π a real matri and Dz = D D D3 be an LMI region. All the eigenvalues of Π are in LMI region Dz if eists a symmetric matri Ψ such that we have the followings LMI h( ΠΨ + ΨΠ ( < ΨΠ ΠΨ h( ΠΨ + ΨΠ rψ ( < qψ + ΨΠ rψ (5 (6 αψ + ΠΨ + ΨΠ < (7 In the net section, we will study the problem of H control with D-stability constraint for uncertain continuous-time repetitive systems which is used to analyze the system stability and to prove the convergence of the tracing error. he synthesis of this control law will be based on the optimization problem under LMI constraints. hα (* (* (* (* (* α6 hα (* (* (* (* hh H ε I (* (* (* < H hh ε I (* (* α5 ε I (* α5 ε I rπ (* (* (* q A Bu r (* (* Π + Π + Γ Π < H ε I (* α5 ε I απ + α (* (* ε H ε I (* < α5 ε I ( ( (3 III. MAIN RESULS his section is devoted to developing the robust H control based on a repetitive control for uncertain system ( and it is desired that the poles of the closed-loop system remain in region D of the comple plane. z Consider the following two-dimensional Lyapunov function V = V + V = P + e Qe (8,, P > and Q > are a symmetrical matrices. he H disturbance attenuation holds if there eist a scalar γ > such that the Hamiltonian satisfies ( + γ ( H ( τ = V τ e e d d τ < (9 A sufficient eistence conditions solutions must be satisfied and the problem of control law synthesis is solved by the following theorem: heorem. : For the uncertain system ( and a given constantγ >, if there eists two symmetric matrices Π >, Π > and two matrices Γ, Γ and a scalar ε > satisfying the followings LMI: Π ( ( ( ( ( ( ( ( α α ( ( ( ( ( ( ( α3 ΓBu Π ( ( ( ( ( ( Bd γ I ( ( ( ( ( α4 Π ( ( ( ( < Π ( ( ( I I ( ( εh ε I ( α5 EBuΓ EBd ε I ( α = Π C + Γ Du α = Π A + AΠ + Γ B + B Γ α 3 = Γ Du Π α 4 = CΠ + Du Γ α5 = EAΠ + EBuΓ α = Π A AΠ + Γ B B Γ u u 6 u u (4 then the system is generalized quadratically D-stable with H performance γ. After resolution of the LMI (-3, the stabilization gains are given by G = Γ. Π, G = Γ. Π (5 Proof. : he associated increment with Lyapunov function is defined by V = Vɺ + V,, = ɺ P + P ɺ (6 + e Qe e Qe Equation (9 can be set in the following form H = e [ Χ ] e < (7 d d Χ ( ( Χ = ( Χ Χ < Bd P γ I (8

4 and Χ = ( A + Bu G P( A + Bu G + ( C + Du G Q( C + Du G Χ = ( Bu G P + ( Du G I Q( C + Du G (9 Χ = ( Du G I Q( Du G I + I Q Let W A + B G B G, = C + D ug D ug I, u u = C + D ug Y =, Z [ I ] B d =, U =. P V =, Q R = Q, S = Q he inequality (8 becomes in the following epression (* W V + VW + R + Y RY + Z Z S Χ = < U V γ I (9 Lemma is used as many times as necessary and (9 can be rewritten by R (* (* (* (* R W V VW S (* (* (* + Χ = U V γ I (* (* < RY R (* Z I (3 After replacing the variables with their epressions in (3, we get the following LMI Q (* (* (* (* (* (* λ λ (* (* (* (* (* λ3 λ4 Q (* (* (* (* Χ = Bd P γ I (* (* (* < λ5 Q (* (* Q (* I I λ = ( C + DuG Q λ = ( A + Bu G P + P( A + Bu G λ3 = ( DuG I Q λ4 = ( Bu G P λ5 = Q( C + DuG (3 (3 Pre-multiply and post-multiply (3, respectively, by {,,,,,,, } diag Q Q P Q I Q Q I the LMI becomes and its transpose. hus, Q (* (* (* (* (* (* δ δ (* (* (* (* (* δ3 δ4 Q (* (* (* (* Χ = B (* (* (* d γ I < (33 δ5 Q (* (* Q (* I I Let δ = P ( C + DuG δ = P ( A + B G + ( A + B G P δ 3 = Q ( DuG I δ 4 = Q ( Bu G δ 5 = ( C + DuG P u u Π = P Π = Q Γ = G Π Γ = G Π H = E = E Π + E Γ E Γ E [ H ] [ ] A Bu Bu Bd (34 (35 Let Σ be the matri that we consider X for the nominal system. Now, it suffices to apply theorem and replace the matri Ψ by Acl = A + Bu G and choose Π = Π. hα α ( hα < 6 rπ (* < q A Bu r Π + Π + Γ Π α u u (36 (37 Π + Π A + A Π + Γ B + B Γ < (38 Let α = Π A + A Π + Γ B + B Γ α = Π A A Π + Γ B B Γ u u 6 u u hα (* Σ = α6 hα hh H H = H hh E = diag( EAΠ + EBuΓ, EAΠ + EBuΓ (39 (4

5 rπ Σ = H H = E = E Π + E Γ qπ + Π A + Γ Bu rπ A Bu (* (4 ( d t π =.5sin t (48 hus, subject to constraints D z we can choose h =, q =, r = 5 and α =. Σ = απ + Π A + AΠ + Γ B + B Γ H3 = H E3 = EAΠ + EBuΓ 3 u u (4 Applying Lemma and Lemma, inequalities (33, can be given in the form Σ H E (* (* (* ε I ε I < inde =,,,3. (43 Equation (43 will be pre-multiplying and post-multiplying, diag I, ε I, I and its transpose. herefore, respectively, by { } the LMI becomes Σ ε H E (* (* ε I (* < ε I (44 After replacing and rearrange correspondent s terms, we get LMI (-3. his completes the proof. Fig. : Simulation results (r(t/y(t, e(t, u(t for the nominal system IV. EXAMPLE In this section, we give an eample to demonstrate the effectiveness of the proposed approach. Consider the following nominal linear system: 3. ɺ ( t = ( t u( t d( t y( t = [ 4 ] ( t + u( t (45 he uncertain matrices are described by: ( ( ep(,sin( F t = diag t t (46 H =., E A =,.5 E Bu =,. E Bd =. he periodic reference trajectory and the eternal disturbance applied to the system has been defined, respectively, by the following functions: π 4π r ( t = sin t +.5sin t (47 Fig. : Simulation results (r(t/y(t, e(t, u(t for the uncertain system For the nominal system, by using heorem, the gains of D controller and parameters of the robust repetitive control are:

6 [ ] [ ] G = , G =.65 Grob = , Grep =.98 (49 Simulation results (reference signal/output, tracing error and control input in Fig. show that the system is stable in closed-loop and enters the steady state in the third period. For the uncertain system, the gains of D controller and parameters of the robust repetitive control are: [ ] [ ] G = , G =.53 Grob =.4.53, Grep =.77 (5 Simulation results are shown in Fig.. It easy to remar that the system is robustly stable for the periodic uncertainties and it enters into the steady state in the seventh period. V. CONCLUSION his paper has as objective to study the problem of H-infinity control with D-stability constraint for uncertain continuoustime repetitive systems with eternal disturbances and design a control law, such that the closed-loop poles are placed within a particular region of the comple plane for all admissible uncertainties. Firstly, an equivalence between a twodimensional control system and a repetitive control scheme such that study of convergence and stability properties have been proved. By analyzing these properties, all of the obtained conditions are formulated in the form of linear matri inequalities and solutions gives the agreed controller gains. Finally, the performance of the proposed control law was tested and simulated on an eample and results are competitive in term of robustness and convergence. Repetitive control is no different from other control laws. It has its advantages, its disadvantages, its problems of robustness and applicability. However, it remains recommended for processes that wor periodically or repetitively. Authors intend to continue research on this problem and the etension of obtained results to other class parameter uncertainty is actually under study. REFERENCES [] B. Zhang, D. Wang, K. Zhou, and Y. 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