Robust H PID Controller Design Via LMI Solution of Dissipative Integral Backstepping with State Feedback Synthesis

Transcription

1 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 4 Endra Joelianto Bandng Institte of echnology Indonesia 1. Introdction PID controller has been extensively sed in indstries since 1940s and still the most often implemented controller today. he PID controller can be fond in many application areas: petrolem processing, steam generation, polymer processing, chemical indstries, robotics, nmanned aerial vehicles (UAVs) and many more. he algorithm of PID controller is a simple, single eqation relating proportional, integral and derivative parameters. Nonetheless, these provide good control performance for many different processes. his flexibility is achieved throgh three adjstable parameters of which vales can be selected to modify the behavior of the closed loop system. A convenient featre of the PID controller is its compatibility with enhancement that provides higher capabilities with the same basic algorithm. herefore the performance of a basic PID controller can be improved throgh jdicios selection of these three vales. Many tning methods are available in the literatre, among with the most poplar method the Ziegler-Nichols (Z-N) method proposed in 194 (Ziegler & Nichols, 194). A drawback of many of those tning rles is that sch rles do not consider load distrbance, model ncertainty, measrement noise, and set-point response simltaneosly. In general, a tning for high performance control is always accompanied by low robstness (Shinskey, 1996). Difficlties arise when the plant dynamics are complex and poorly modeled or, specifications are particlarly stringent. Even if a soltion is eventally fond, the process is likely to be expensive in terms of design time. Varieties of new methods have been proposed to improve the PID controller design, sch as analytical tning (Boyd & Barrat, 1991; Hwang & Chang, 1987), optimization based (Wong & Seborg, 1988; Boyd & Barrat, 1991; Astrom & Hagglnd, 1995), gain and phase margin (Astrom & Hagglnd, 1995; Fng et al., 1998). Frther improvement of the PID controller is soght by applying advanced control designs (Ge et al., 00; Hara et al., 006; Wang et al., 007; Goncalves et al., 008). In order to design with robst control theory, the PID controller is expressed as a state feedback control law problem that can then be solved by sing any fll state feedback robst control synthesis, sch as Garanteed Cost Design sing Qadratic Bond (Petersen et al., 000), H synthesis (Green & Limebeer, 1995; Zho & Doyle, 1998), Qadratic Dissipative Linear Systems (Yliar et al., 1997) and so forth. he PID parameters selection by

2 70 Robst Control, heory and Applications transforming into state feedback sing linear qadratic method was first proposed by Williamson and Moore in (Williamson & Moore, 1971). Preliminary applications were investigated in (Joelianto & omy, 003) followed the work in (Joelianto et al., 008) by extending the method in (Williamson & Moore, 1971) to H synthesis with dissipative integral backstepping. Reslts showed that the robst H PID controllers prodce good tracking responses withot overshoot, good load distrbance responses, and minimize the effect of plant ncertainties cased by non-linearity of the controlled systems. Althogh any robst control designs can be implemented, in this paper, the investigation is focsed on the theory of parameter selection of the PID controller based on the soltion of robst H which is extended with fll state dissipative control synthesis and integral backstepping method sing an algebraic Riccati ineqality (ARI). his paper also provides detailed derivations and improved conditions stated in the previos paper (Joelianto & omy, 003) and (Joelianto et al., 008). In this case, the selection is made via control system optimization in robst control design by sing linear matrix ineqality (LMI) (Boyd et al., 1994; Gahinet & Apkarian, 1994). LMI is a convex optimization problem which offers a nmerically tractable soltion to deal with control problems that may have no analytical soltion. Hence, redcing a control design problem to an LMI can be considered as a practical soltion to this problem (Boyd et al., 1994). Solving robst control problems by redcing to LMI problems has become a widely accepted techniqe (Balakrishnan & Wang, 000). General mlti objectives control problems, sch as H and H performance, peak to peak gain, passivity, regional pole placement and robst reglation are notoriosly difficlt, bt these can be solved by formlating the problems into linear matrix ineqalities (LMIs) (Boyd et al., 1994; Scherer et al., 1997)). he objective of this paper is to propose a parameter selection techniqe of PID controller within the framework of robst control theory with linear matrix ineqalities. his is an alternative method to optimize the adjstment of a PID controller to achieve the performance limits and to determine the existence of satisfactory controllers by only sing two design parameters instead of three well known parameters in the PID controller. By sing optimization method, an absolte scale of merits sbject to any designs can be measred. he advantage of the proposed techniqe is implementing an otpt feedback control (PID controller) by taking the simplicity in the fll state feedback design. he proposed techniqe can be applied either to a single-inpt-single-otpt (SISO) or to a mlti-inpts-mlti-otpts (MIMO) PID controller. he paper is organised as follows. Section describes the formlation of the PID controller in the fll state feedback representation. In section 3, the synthesis of H dissipative integral backstepping is applied to the PID controller sing two design parameters. his section also provides a derivation of the algebraic Riccati ineqality (ARI) formlation for the robst control from the dissipative integral backstepping synthesis. Section 4 illstrates an application of the robst PID controller for time delay ncertainties compensation in a network control system problem. Section 5 provides some conclsions.. State feedback representation of PID controller In order to design with robst control theory, the PID controller is expressed as a fll state feedback control law. Consider a single inpt single otpt linear time invariant plant described by the linear differential eqation

3 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 71 xt () = Axt () + Bt () yt () = Cxt () with some ncertainties in the plant which will be explained later. Here, the states x R 1 are the soltion of (1), the control signal R is assmed to be the otpt of a PID 1 controller with inpt y R. he PID controller for reglator problem is of the form t d t () = K y()() tdt + Ky() t + K y() t () dt which is an otpt feedback control system and K1 = Kp / i, K = Kp, K3 = Kpd of which K p, i and d denote proportional gain, time integral and time derivative of the well known PID controller respectively. he strctre in eqation () is known as the standard PID controller (Astrom & Hagglnd, 1995). he control law () is expressed as a state feedback law sing (1) by differentiating the plant otpt y as follows y = Cx y = C Ax+ C B y = C A x+ C AB + C B his means that the derivative of the control signal () may be written as 3 1 (1 KCB 3 ) ( KC A + KC A+ KC ) x ( KC 3 AB + KCB ) = 0 (3) Using the notation ˆK as a normalization of K, this can be written in more compact form or ˆK ˆ [ ˆ ˆ ˆ 1 K = K K K ] = (1 KCB) [ K K K ] (4) = ck where c is a scalar. his control law is then given by ˆ[ ( ) ] = K C A C A C x+ (1) Kˆ[0 B C B A C ] (5) Denote ˆ[ ( ) ] Kx = K C A C A C and ˆ[0 ] K = K B C B A C, the block diagram of the control law (5) is shown in Fig. 1. In the state feedback representation, it can be seen that the PID controller has interesting featres. It has state feedback in the pper loop and pre integrator backstepping in the lower loop. By means of the internal model principle (IMP) (Francis & Wonham, 1976; Joelianto & Williamson, 009), the integrator also garantees that the PID controller will give zero tracking error for a step reference signal. Eqation (5) represents an otpt feedback law with constrained state feedback. hat is, the control signal () may be written as n where = K x (6) a a a a =, x xa =

4 7 Robst Control, heory and Applications ˆ [ ( ) ] [0 ] Ka = K C A C A C B C B A C Arranging the eqation and eliminating the transpose lead to C 0 ˆ Ka = K CA CB CA CAB he agmented system eqation is obtained from (1) and (7) as follows = ˆK Γ (7) where x = A x + B (8) a a a a a A B = 0 0 ; A a B a 0 = 1 K x + = Ax + B x x C y + K Fig. 1. Block diagram of state space representation of PID controller Eqation (6), (7) and (8) show that the PID controller can be viewed as a state variable feedback law for the original system agmented with an integrator at its inpt. he agmented formlation also shows the same strctre known as the integral backstepping method (Krstic et al., 1995) with one pre integrator. Hence, the selection of the parameters of the PID controller (6) via fll state feedback gain is inherently an integral backstepping control problems. he problem of the parameters selection of the PID controller becomes an optimal problem once a performance index of the agmented system (8) is defined. he parameters of the PID controller are then obtained by solving eqation (7) that reqires the inversion of the matrix Γ. Since Γ is, in general, not a sqare matrix, a nmerical method shold be sed to obtain the inverse.

5 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 73 For the sake of simplicity, the problem has been set-p in a single-inpt-single-otpt (SISO) case. he extension of the method to a mlti-inpts-mlti-otpts (MIMO) case is m straighforward. In MIMO PID controller, the control signal has dimension m, R is p assmed to be the otpt of a PID controller with inpt has dimension p, y R. he parameters of the PID controller K 1, K, and K3 will be sqare matrices with appropriate dimension. 3. H dissipative integral backstepping synthesis he backstepping method developed by (Krstic et al., 1995) has received considerable attention and has become a well known method for control system designs in the last decade. he backstepping design is a recrsive algorithm that steps back toward the control inpt by means of integrations. In nonlinear control system designs, backstepping can be sed to force a nonlinear system to behave like a linear system in a new set of coordinates with flexibility to avoid cancellation of sefl nonlinearities and to focs on the objectives of stabilization and tracking. Here, the paper combines the advantage of the backstepping method, dissipative control and H optimal control for the case of parameters selection of the PID controller to develop a new robst PID controller design. Consider the single inpt single otpt linear time invariant plant in standard form sed in H performance by the state space eqation n xt () = Axt () + Bwt () + Bt (), x(0) = x 1 0 zt () = Cxt () + D wt () + D t () yt () = Cxt () + D wt () + D t () 1 where x R denotes the state vector, R is the control inpt, w R is an external inpt and represents driving signals that generate reference signals, distrbances, and 1 m measrement noise, y R is the plant otpt, and z R is a vector of otpt signals related to the performance of the control system. Definition 1. A system is dissipative (Yliar et al., 1998) with respect to spply rate rzt ((), wt ()) for each n initial condition x 0 if there exists a storage fnction V, V : R R + satisfies the ineqality 1 p (9) t1 Vxt ( ( )) + rzt ( ( ), wt ( )) dt Vxt ( ( )), ( t1, t0) R + n, x0 R (10) 0 1 t0 and t0 t1 and all trajectories ( x, y, z ) which satisfies (9). he spply rate fnction rzt ((), wt ()) shold be interpreted as the spply delivered to the t system. If in the interval [ t0, t 1] the integral 1 rzt ((), wt ()) dt is positive then work has been t0 done to the system. Otherwise work is done by the system. he spply rate determines not only the dissipativity of the system bt also the reqired performance index of the control system. he storage fnction V generalizes the notion of an energy fnction for a dissipative system. he fnction characterizes the change of internal storage Vxt ( ( 1)) Vxt ( ( 0)) in any interval [ t0, t 1], and ensres that the change will never exceed the amont of the spply into

6 74 Robst Control, heory and Applications the system. he dissipative method provides a nifying tool as index performances of control systems can be expressed in a general spply rate by selecting vales of the spply rate parameters. he general qadratic spply rate fnction (Hill & Moylan, 1977) is given by the following eqation where Q and R are symmetric matrices and 1 rzw (, ) = ( wqw+ wsz+ zrz) (11) Q( x) = Q+ SD ( x) + D ( x) S + D ( x) RD ( x) n sch that Qx ( ) > 0 for x R and R 0 and inf n { σ min( Qx ( ))} = k> 0. his general x R spply rate represents general problems in control system designs by proper selection of matrices Q, R and S (Hill & Moylan, 1977; Yliar et al., 1997): finite gain (H ) performance ( Q= γ I, S = 0 and R = I); passivity ( Q= R= 0 and S= I ); and mixed H - positive real performance ( Q= θγ I, R= θ I and S = (1 θ ) I for θ [0,1] ). For the PID control problem in the robst control framework, the plant ( Σ ) is given by the state space eqation xt () = Axt () + Bwt () + Bt (), x(0) = x Σ = Cxt 1 () zt () = D1() t 1 0 with D 11 = 0 and γ > 0 with the qadratic spply rate fnction for H performance (1) 1 rzw (, ) = ( γ ww zz) (13) Next the plant ( Σ ) is added with integral backstepping on the control inpt as follows xt () = Axt () + Bwt 1 () + Bt () () t = () t (14) a Cxt 1 () zt () = D1t () ρa() t where ρ is the parameter of the integral backstepping which act on the derivative of the control signal t (). In eqation (14), the parameter ρ > 0 is a tning parameter of the PID controller in the state space representation to determine the rate of the control signal. Note that the standard PID controller in the state feedback representation in the eqations (6), (7) and (8) corresponds to the integral backstepping parameter ρ = 1. Note that, in this design the gains of the PID controller are replaced by two new design parameters namely γ and ρ which correspond to the robstness of the closed loop system and the control effort. he state space representation of the plant with an integrator backstepping in eqation (14) can then be written in the agmented form as follows

7 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 75 xt () A B xt () B1 0 wt () a() t t () = 0 0 t () C1 0 0 xt () zt () = 0 D a() t t () 0 0 ρ he compact form of the agmented plant ( Σ ) is given by a (15) where x () t = A x () t + B w() t + B (); t x (0) = x a a a w a a a a0 zt () = Cx() t + D wt () + D () t a a 1a a a (16) x A B xa =, Aa = 0 0, B w B1 = 0, Now consider the fll state gain feedback of the form B a 0 = 1, C C 0 1 a = 0 D1, D a = 0 ρ () t = K x () t (17) a a a he objective is then to find the gain feedback K a which stabilizes the agmented plant ( Σ a ) with respect to the dissipative fnction V in (10) by a parameter selection of the qadratic spply rate (11) for a particlar control performance. Fig.. shows the system description of the agmented system of the plant and the integral backstepping with the state feedback control law. z y x a Σ a w a = a K a x a Fig.. System description of the agmented system

8 76 Robst Control, heory and Applications he following theorem gives the existence condition and the formla of the stabilizing gain feedback K a. heorem. Given γ > 0 and ρ > 0. If there exists X = X > 0 of the following Algebraic Riccati Ineqality A B A BB X X X C1C1 0 + ρ γ X+ 0 0 B < 0 D1D1 hen the fll state feedback gain (18) leads to Σ <γ [ ] K = ρ B X = ρ 0 1 X (19) Proof. Consider the standard system (9) with the fll state feedback gain and the closed loop system a a t () = Kxt () xt () = Axt () + Bwt (), x(0) = x zt () = C xt () + D wt () 1 0 where D 11 = 0, A = A+ BK, C = C1 + D1K is strictly dissipative with respect to the qadratic spply rate (11) sch that the matrix A is asymptotically stable. his implies that the related system 1 11 xt () = Axt () + Bwt (), x(0) = x zt () = Cxt () where A 1/ = A B1Q SC, B 1 = B1Q and 1 1/ C 1 = ( S Q S R) C has H norm strictly less than 1, which implies there exits a matrix X > 0 solving the following Algebraic Riccati Ineqality (ARI) (Petersen et al. 1991) A X XA XB B X C C (0) < 0 In terms of the parameter of the original system, this can be written as ( A ) X + XA + [ XB ( C ) S ] Q [ B X SC ] ( C ) RC < 0 (1) Define the fll state feedback gain By inserting (( ) K = E B B Q SD X + D RC ()

9 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 77, , , , 1 1 A = A+ B K C = C + D K S = S+ D R Q= Q+ SD + D S + D RD R = S Q S R E = D RD B= B B Q SD D= I D E D R into (1), completing the sqares and removing the gain K give the following ARI X( A BE D RC B QSC ) + ( A BE D RC B QSC ) X XBE ( B BQ B ) X+ C DRDC < 0 (3) Using the reslts (Scherer, 1990), if there exists X > 0 satisfies (3) then K given by () is stabilizing sch that the closed loop system A = A+ BK is asymptotically stable. Now consider the agmented plant with integral backstepping in (16). In this case, 1 [ 0 0 0] D a =. Note that D ac a = 0 and D 1 a = 0. he H performance is satisfied by setting the qadratic spply rate (11) as follow: a a a a a 1 a( a a) a S= 0, R= R= I, E = D RD = D D, B= B, D= I D D D D Inserting the setting to the ARI (3) yields 1 1 a a a a a a w a a a a a a a w a 1 1 a a a a w w 1 1 a a a a a a a a a a XA ( B( D D ) D IC BQ 0 C) + ( A B ( D D ) D IC B Q 0 C ) X XB ( ( D D ) B BQ B ) X+ + ( C ( I D ( D D ) D ) x ( I D ( D D ) D ) C ) < 0 he eqation can then be written in compact form a a a a w w a a XA + A X X( ρ B B γ B B ) X + C C < 0 (4) this gives (18). he fll state feedback gain is then fond by inserting the setting into () (( ) ) 1 1 a = a w a a a K E B B Q SD X D RC that gives Σ < γ (Yliar et al., 1998; Scherer & Weiland, 1999). his completes the proof. he relation of the ARI soltion (8) to the ARE soltion is shown in the following. Let the transfer fnction of the plant (9) is represented by and assme the following conditions hold: (A1). ( A, B, C ) is stabilizable and detectable (A). D = 0 zs () P11() s P1() s ws () y() s = P () s P () s () s 1

10 78 Robst Control, heory and Applications (A3). D 1 has fll colmn rank, D 1 has fll row rank (A4). P 1 () s and P 1 () s have no invariant zero on the imaginary axis From (Gahinet & Apkarian, 1994), eqivalently the Algebraic Riccati Eqation (ARE) given by the formla = X( A BE D RC B QSC ) + ( A BE D RC B QSC ) X XBE ( B BQ B ) X C DRDC 0 has a (niqe) soltion X 0, sch that (5) A+ B K + B Q [ B X S( C + D K)] is asymptotically stable. he characterization of feasible γ sing the Algebraic Riccati Ineqality (ARI) in (4) and ARE in (5) is immediately where the soltion of ARE ( X ) and ARI ( X 0 ) satisfy 0 X < X0, X0 = X0 > 0 (Gahinet & Apkarian, 1994). he Algebraic Riccati Ineqality (4) by Schr complement implies AaX + XAa + CaCa XBa XB w BX a ρ I 0 < 0 BwX 0 γ I her problem is then to find X > 0 sch that the LMI given in (6) holds. he LMI problem can be solved sing the method (Gahinet & Apkarian, 1994) which implies the soltion of the ARI (18) (Li & He, 006). he parameters of the PID controller which are designed by sing H dissipative integral backstepping can then be fond by sing the following algorithm: 1. Select ρ> 0. Select γ > 0 3. Find X 0 > 0 by solving LMI in (6) 4. Find K a sing (19) 5. Find ˆK sing (7) 6. Compte K 1, K and K 3 sing (4) 7. Apply in the PID controller () 8. If it is needed to achieve γ minimm, repeat step and 3 ntil γ =γ min then follows the next step (6) 4. Delay time ncertainties compensation Consider the plant given by a first order system with delay time which is common assmption in indstrial process control and frther assme that the delay time ncertainties belongs to an a priori known interval 1 Ls Ys () = e Us (), L [ a, b] (7) τ s + 1 he example is taken from (Joelianto et al., 008) which represents a problem in indstrial process control de to the implementation of indstrial digital data commnication via

11 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 79 ethernet networks with fieldbs topology from the controller to the sensor and the actator (Hops et al., 004; Jones, 006, Joelianto & Hosana, 009). In order to write in the state space representation, the delay time is approximated by sing the first order Pade approximation 1 ds + 1 Ys () = Us (), d = L/ (8) τ s + 1 ds + 1 In this case, the vales of τ and d are assmed as follows: τ = 1 s and d nom = 3 s. hese pose a difficlt problem as the ratio between the delay time and the time constant is greater than one ( ( d / τ ) > 1). he delay time ncertainties are assmed in the interval d [,4]. he delay time ncertainty is separated from its nominal vale by sing linear fractional transformation (LF) that shows a feedback connection between the nominal and the ncertainty block. δ θ yθ d y Fig. 3. Separation of nominal and ncertainty sing LF he delay time ncertainties can then be represented as d = α d nom +βδ, 1 <δ< d = F, δ β α After simplification, the delay time ncertainties of any known ranges have a linear fractional transformation (LF) representation as shown in the following figre. δ θ yθ G tot y Fig. 4. First order system with delay time ncertainty

12 80 Robst Control, heory and Applications he representation of the plant agmented with the ncertainty is he corresponding matrices in (9) are A B B G () s = = C D D tot 1 Ax Bx Cx D x C D1 D (9) Ax11 0 Bx11 Bx Ax = 1 1, Bx = 0 1, C x Cx11 0 Dx11 Dx1 = 0 1, Dx = 0 0 In order to incorporate the integral backstepping design, the plant is then agmented with an integrator as follows Ax11 0 Bx11 A B Aa = = B C w Bx 11 B1 = = 0 0, 0 B a 0 0 = 0 = I, 1 C 0 x11 a = 0 Dx1 D a = 0 ρ he problem is then to find the soltion X > 0 and γ > 0 of ARI (18) and to compte the fll state feedback gain given by, xt () () ([ 0 1] ) () a t = Kaxa t = ρ X t () which is stabilizing and leads to the infinity norm Σ < γ. he state space representation for the nominal system is given by A nom = 1 0, B nom 1 = 0, C = [ ] nom

13 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 81 In this representation, the performance of the closed loop system will be garanteed for the specified delay time range with fast transient response (z). he fll state feedback gain of the PID controller is given by the following eqation ˆ K1 K 1 1 K ( 1 K3[ ] ) Kˆ = 0 K ˆ 3 K3 For different γ, the PID parameters and transient performances, sch as: settling time ( s ) and rise time ( r ) are calclated by sing LMI (6) and presented in able 1. For different ρ bt fixed γ, the parameters are shown in able. As comparison, the PID parameters are also compted by sing the standard H performance obtained by solving ARE in (5). he reslts are shown able 3 and able 4 for different γ and different ρ respectively. It can be seen from able 1 and that there is no clear pattern either in the settling time or the rise time. Only able 1 shows that decreasing γ decreases the vale of the three parameters. On the other hand, the calclation sing ARE shows that the settling time and the rise time are decreased by redcing γ or ρ. able 3 shows the same reslt with the able 1 when the vale of γ is decreased. γ ρ K p K i K d r (s) s 5% (s) able 1. Parameters and transient response of PID for different γ with LMI γ ρ K p K i K d r (s) s 5% (s) able. Parameters and transient response of PID for different ρ with LMI

14 8 Robst Control, heory and Applications γ ρ K p K i K d r (s) s 5% (s) able 3. Parameters and transient response of PID for different γ with ARE γ ρ K p K i K d r (s) s 5% (s) able 4. Parameters and transient response of PID for different ρ with ARE Fig. 5. ransient response for different γ sing LMI

15 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 83 Fig. 6. ransient response for different ρ sing LMI Fig. 7. Nyqist plot γ = 0.48 and ρ = 1 sing LMI

16 84 Robst Control, heory and Applications Fig. 8. Nyqist plot γ = and ρ = 0.66 sing LMI Fig. 9. ransient response for different d sing LMI

17 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 85 Fig. 10. ransient response for different bigger d sing LMI he simlation reslts are shown in Figre 5 and 6 for LMI, with γ and ρ are denoted by g and r respectively in the figre. he LMI method leads to faster transient response compared to the ARE method for all vales of γ and ρ. Nyqist plots in Figre 7 and 8 show that the LMI method has small gain margin. In general, it also holds for phase margin except at γ = and ρ = 1.5 where LMI has bigger phase margin. In order to test the robstness to the specified delay time ncertainties, the obtained robst PID controller with parameter γ =0.1 and ρ = 1 is tested by pertrbing the delay time in the range vale of d [1, 4]. he reslts of sing LMI are shown in Figre 9 and 10 respectively. he LMI method yields faster transient responses where it tends to oscillate at bigger time delay. With the same parameters γ and ρ, the PID controller is sbjected to bigger delay time than the design specification. he LMI method can handle the ratio of delay time and time constant L / τ 1 s while the ARE method has bigger ratio L / τ 43 s. In smmary, simlation reslts showed that LMI method prodced fast transient response of the closed loop system with no overshoot and the capability in handling ncertainties. If the range of the ncertainties is known, the stability and the performance of the closed loop system will be garanteed. 5. Conclsion he paper has presented a model based method to select the optimm setting of the PID controller sing robst H dissipative integral backstepping method with state feedback synthesis. he state feedback gain is fond by sing LMI soltion of Algebraic Riccati Ineqality (ARI). he paper also derives the synthesis of the state feedback gain of robst H dissipative integral backstepping method. he parameters of the PID controller are

18 86 Robst Control, heory and Applications calclated by sing two new parameters which correspond to the infinity norm and the weighting of the control signal of the closed loop system. he LMI method will garantee the stability and the performance of the closed loop system if the range of the ncertainties is inclded in the LF representation of the model. he LF representation in the design can also be extended to inclde plant ncertainties, mltiplicative pertrbation, pole clstering, etc. Hence, the problem will be considered as mlti objectives LMI based robst H PID controller problem. he proposed approach can be directly extended for MIMO control problem with MIMO PID controller. 6. References Astrom, K.J. & Hagglnd,. (1995). PID Controllers: heory, Design, and ning, second ed., Instrment Society of America, ISBN , Research riangle Park, North Carolina - USA Boyd, S.P. & Barrat, C.H. (1991). Linear Controller Design: Limits of Performance, Prentice Hall Inc., ISBN X, New Jersey Boyd, S.; El Ghaoi, L., Feron, E. & Balakrishnan, V. (1994). Linear Matrix Ineqalities in System and Control heory, SIAM Stdies 15, ISBN , Philadelphia Balakrishnan, V. & Wang, F. (000). Semidefinite programming in systems and control, In: Handbook on Semidefinite Programming, Wolkowics, H; Saigal, R. & Vandenberghe, L. (Ed.), pp , Klwer Academic Pb., ISBN , Boston Francis, B.A. & Wonham, W.M. (1976). he internal model principle of control theory, Atomatica, Vol. 1, pp , ISSN Fng, H.W.; Wang, Q.G. & Lee,.H. (1998). PI tning in terms of gain and phase margins, Atomatica, Vol. 34, pp , ISSN Gahinet, P. & Apkarian, P. (1994). A linear matrix ineqality approach to H control, Inter. Jornal of Robst Nonlinear Control, Vol. 4, pp , ISSN Ge, M.; Chi, M.S. & Wang, Q.G. (00). Robst PID controller design via LMI approach, Jornal of Process Control, Vol. 1, pp. 3-13, ISSN Green, M. & Limebeer, D.J. (1995). Linear Robst Control, Englewood Cliffs, Prentice Hall Inc., ISBN , New Jersey Goncalves, E.N.; Palhares, R.M. & akahashi, R.H.C. (008). A novel approach for H /H robst PID synthesis for ncertain systems, Jornal of Process Control, Vol. 18, pp. 19-6, ISSN Hara, S.; Iwasaki,. & Shiokata, D. (006). Robst PID control sing generalized KYP synthesis, IEEE Control Systems Magazine, Feb., pp , ISSN Hill, D.J. & Moylan, P.J. (1977). Stability reslts for nonlinear feedback systems, Atomatica, Vol. 13, pp , ISSN Hops, J.; Swing, B., Phelps, B., Sdweeks, B., Pane, J. & Kinslow, J. (004). Non-deterministic DU behavior dring fnctional testing of high speed serial bsses: challenges and soltions, Proceedings International est Conference, ISBN , 6-8 Oct. 004, IEEE, Charlotte, NC, USA Hwang, S.H. & Chang, H.C. (1987). A theoritical examination of closed-loop properties and tning methods of single-loop PI controllers, Chemical Engineering Science, Vol. 4, pp , ISSN

19 Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis 87 Joelianto, E. & ommy. (003). A robst DC to DC bckboost converter sing PID H backstepping controller, Proceedings of Int. Confer. on Power Electronics and Drive Systems (PEDS), pp , ISBN , 17-0 Nov. 003, Singapore Joelianto, E.; Starto, H.Y. & Wicaksono, A. (008). Compensation of delay time ncertainties indstrial control ethernet networks sing LMI based robst H PID controller, Proceedings of 5 th Int. Confer. Wireless Optical Comm. Networks (WOCN), pp. 1-5, ISBN , 5-7 May 008, Srabaya - Indonesia Joelianto, E. & Hosana (009). Loop-back action latency performance of an indstrial data commnication protocol on a PLC ethernet network, Internetworking Indonesia Jornal, Vol. I, No.1, pp , ISSN Joelianto, E. & Williamson, D. (009). ransient response improvement of feedback control systems sing hybrid reference control, International Jornal of Control, Vol. 81, No. 10, pp , ISSN print/issn online Jones, M. (006). Designning for real time embedded ethernet, he Indstrial Ethernet Book, Vol. 35, pp Krstic, M.; Kanellakopolos, I. & Kokotovic, P. (1995). Nonlinear and Adaptive Control Design, John Wiley and Sons Inc., ISBN , USA Li, K.Z. & He, R. (006). A simple derivation of ARE soltions to the standard H control problem based on LMI soltion, Systems & Control Letters, Vol. 5, pp , ISSN Petersen, I.R.; Anderson, B.D.O. & Jonkheere, E. (1991). A first principles soltion to the nonsinglar H control problem, Inter. Jornal of Robst and Nonlinear Control, Vol. 1, pp , ISSN Petersen, I.R.; Ugrinovskii, V.A. & Savkin, A.V. (000). Robst Control Design sing H Methods, Springer, ISBN , London Scherer, C. (1990). he Riccati Ineqality and State-Space H -Optimal Control, PhD Dissertation, Bayerischen Jlis Maximilans, Universitat Wrzbrg, Wrzbrg Scherer, C.; Gahinet, P. & Chilali, M. (1997). Mltiobjective otpt-feedback control via LMI optimization, IEEE rans. on Atomatic Control, Vol. 4, pp , ISSN Scherer, C. & Weiland, S. (1999). Linear Matrix Ineqalities in Control, Lectre Notes DISC Corse, version.0. Shinskey, F.G. (1996). Process Control Systems: Application, Design and ning, forth ed., McGraw-Hill, ISBN , Boston Wang, Q.G.; Lin, C., Ye, Z., Wen, G., He, Y. & Hang, C.C. (007). A qasi-lmi approach to compting stabilizing parameter ranges of mlti-loop PID controllers, Jornal of Process Control, Vol. 17, pp. 59-7, ISSN Williamson, D. & Moore, J.B. (1971). hree term controller parameter selection sing sboptimal reglator theory, IEEE rans. on Atomatic Control, Vol. 16, pp. 8-83, ISSN Wong, S.K.P. & Seborg, D.E. (1988). Control strategy for single-inpt-single-otpt nonlinear systems with time delays, International Jornal of Control, Vol. 48, pp , ISSN print/issn online Yliar, S.; James, M.R. & Helton, J.W. (1998). Dissipative control systems synthesis with fll state feedback, Mathematics of Control, Signals, and Systems, Vol. 11, pp , ISSN print/issn X online

20 88 Robst Control, heory and Applications Yliar, S.; Samydia, Y. & Kadiman, K. (1997). General linear qadratic dissipative otpt feedback control system synthesis, Proceedings of nd Asian Control Confererence (ASCC), pp , ISBN , Seol-Korea, ACPA Zho, K. & Doyle, J.C. (1998). Essential of Robst Control, Prentice Hall Inc., ISBN , USA Ziegler, J.G. & Nichols, N.B. (194). Optimm setting for atomatic controllers, ransactions of he ASME, Vol. 64, pp

21 Robst Control, heory and Applications Edited by Prof. Andrzej Bartoszewicz ISBN Hard cover, 678 pages Pblisher Inech Pblished online 11, April, 011 Pblished in print edition April, 011 he main objective of this monograph is to present a broad range of well worked ot, recent theoretical and application stdies in the field of robst control system analysis and design. he contribtions presented here inclde bt are not limited to robst PID, H-infinity, sliding mode, falt tolerant, fzzy and QF based control systems. hey advance the crrent progress in the field, and motivate and encorage new ideas and soltions in the robst control area. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following: Endra Joelianto (011). Robst H PID Controller Design Via LMI Soltion of Dissipative Integral Backstepping with State Feedback Synthesis, Robst Control, heory and Applications, Prof. Andrzej Bartoszewicz (Ed.), ISBN: , Inech, Available from: Inech Erope University Camps SeP Ri Slavka Kratzeka 83/A Rijeka, Croatia Phone: +385 (51) Fax: +385 (51) Inech China Unit 405, Office Block, Hotel Eqatorial Shanghai No.65, Yan An Road (West), Shanghai, 00040, China Phone: Fax: