CHEAP CONTROL PERFORMANCE LIMITATIONS OF INPUT CONSTRAINED LINEAR SYSTEMS

Copyright 22 IFAC 5th Triennial World Congre, Barcelona, Spain CHEAP CONTROL PERFORMANCE LIMITATIONS OF INPUT CONSTRAINED LINEAR SYSTEMS Tritan Pérez Graham C. Goodwin Maria M. Serón Department of Electrical and Computer Engineering, The Univerity of Newcatle, Autralia. Laboratory for Dynamical Sytem and Signal Proceing (LSD), Univeridad Nacional de Roario, Argentina. Abtract: Limited authority of actuator implie that real control ignal are alway contrained, and in almot all cae, thi produce a degradation in the performance of the ytem. It i thu of practical importance to undertand the fundamental apect of thi performance degradation. In thi paper, we take an initial tep by propoing a way to characterize the performance limitation that arie in cloed-loop table linear ytem due to the contraint on the magnitude of the control ignal. Specifically, we evaluate the cot aociated with the contraint via the L 2 norm of the tracking error of a contrained limiting optimal compenator. Keyword: Performance limitation, cheap control, contrained control ytem.. INTRODUCTION The iue of limitation in control ha given rie to on-going interet ince Bode original work in the 94 (Bode, 945). Thi iue i central to any feedback ytem ince it reveal what can and cannot be achieved given the ytem tructural and dynamical characteritic. The tool for analyzing the limit of performance for uncontrained linear ytem include logarithmic enitivity integral and limiting linear quadratic optimal control. The firt approach ha been extenively developed and reviewed, for example, in (Serón et al., 997), while the econd ha been tudied, for example, by (Qiu and Davion, 993). Thee two apparently independent tool have alo recently been hown to be intimately related (Middleton and Bralavky, 2). In thi paper, we propoe a way to characterize the performance limitation in cloed-loop table linear ytem that arie due to contraint on the magnitude of the control ignal. In our approach, we evaluate the cot aociated with the contraint uing the value of the L 2 norm of the tracking error of a contrained limiting optimal compenator when a unit tep reference i applied. To olve thi problem, we exploit the relationhip between limiting linear quadratic optimal control problem and open loop invere control. We focu on SISO linear ytem. The ret of the paper i organized a follow. In ection 2, we review the o called cheap limiting optimal control problem and the ue of the optimal cot a a meaure of performance limitation. In ection 3, we utilize thi key idea to evaluate the L 2 norm of the tracking error for a tep input in a contrained cheap control compenator. In ection 4, we preent example to illutrate the method. Finally, in ection 5, we draw concluion. 2. CHEAP CONTROL FUNDAMENTAL LIMITATIONS Allowing arbitrarily large control ignal i obviouly impractical. However, the fact that, even under thee condition, the cot function can-

not be reduced to zero expoe the preence of fundamental limitation that are related to the tructure and dynamic of a ytem. Thu, by not retricting the control effort (Cheap control), the cot obtained i a benchmark againt which other more realitic cenario can be judged. In thi ection we review propertie of cheap optimal control for linear ytem. We will ue thi a a bai for the analyi of the contrained cae in ection 3. Conider a linear time-invariant ytem: ẋ(t) =Ax(t)+Bu(t), x R n,u R y(t) =Cx(t), y R,x() = x, which i aumed to be tabilizable and detectable, and the cot functional ( ) J ε = y t (t)y(t)+ε 2 u t (t)u(t) dt. (2) The cheap control problem eek the tate feedback control u that minimize J ε a ε tend to zero. The tructure of the olution to the cheap control problem i better appreciated if we make a change of variable x [η t z t ] t taking the ytem into the normal form or zero dynamic form (Iidori, 995): η = A η + B z ż = A η + A 2 z + B u y = z. In (3), η R m, z = [z,z 2,...,z n m ] t R n m, and the eigenvalue of A are the m zero of the ytem tranfer function H() = C(I A) B.Theη-ubytem i called the zerodynamic ubytem. With the ytem matrice in the form (3), and conidering ε>, the olution P (ε) > can be computed, in the form of a erie in ε, uing the Riccati equation aociated with (2): () (3) A t P (ε)+p (ε)a + C t C ε 2 P (ε)bbt P (ε) =. (4) Once P (ε) i obtained, taking the limit a ε, the cheap control cot i found to be (Serón et al., 999) J lim ε J ε = y 2 (t)dt = η() t P η(), (5) where P i a poitive emi-definite olution of A t P + P A = P B B t P. (6) For implicity, we aume that the zero of (3) (i.e., the eigenvalue of A ) are either all minimum phae or all non-minimum phae. If the zero are all minimum phae, the olution of (6) i P =, and therefore the cot (5) i zero. On the other hand, if the zero are all nonminimum phae then A i Hurwitz and (6) ha a unique poitive definite olution P > (note that (A,B ) i controllable, becaue the ytem (3) i tabilizable and A i Hurwitz). Thu, the cheap control cot (5) i nonzero in thi cae if the initial zero-dynamic tate i different from zero. When ε> i mall, the optimal control olution ha high gain and induce a two-time-cale decompoition in the cloed-loop ytem. To implify the analyi, uppoe, for the moment, that the ytem (3) ha relative degree one, i.e., z = z = y and B i a calar in (3). Then the optimal control i u opt ε = ε [y + Bt P η + O(ε)] (7) O(ε), where lim ε i a nonzero finite real number. ε The cloed-loop ytem i then in the tandard ingular perturbation form: η = A η + B y, εẏ = B (y + B t P η)+o(ε). For a low-fat analyi (Kokotović et al., 986) we et ε = in the econd equation of (8) and obtain the output a a function of the zero-dynamic tate, i.e., y = B t P η. Then, ubtituting in the η-equation, we find that the low ubytem of (8) i the optimal zero-dynamic ubytem (8) η = P A t P η. (9) Thee low dynamic evolve in the ingular hyperplane y + B t P η =, while the fat dynamic repreent the convergence of y + BP t η to zero. In the limit a ε, no matter what the initial condition (η(),z()) i, the tate (η( + ),z( + )) i on thi ingular hyperplane, and evolve inide thi ubpace thereafter. The initial fat repone of the tate i ingular and o i the control that take the tate from the initial condition into the hyperplane (Franci, 979). Once the tate i on the hyperplane, it preent a low evolution given by the dynamic of (9) with y = BP t η. For ytem with higher relative degree, a imilar analyi how that the ingular hyperplane i given by (Saberi and Sannuti, 987) [B t P...] [η t z t ] t =. () For the minimum phae cae thi ingular hyperplane i imply the origin. Cheap Tracking Performance Conider now the problem of regulating the output to a contant etpoint r tarting from zero initial tate. Define the error variable e(t) = y(t) r, η(t) =η(t) η, z i (t) =z i (t) z i, i =2,...,n m, v(t) =u(t) ū, where the bar on the variable denote their teady tate value correponding to the etpoint r. Then the cheap contant etpoint tracking problem i equivalent to the cheap regulator problem decribed above with the error variable replacing the original variable and with cot J lim (e 2 (t)+ε 2 v 2 (t))dt. ε

From (5) we know that the optimal value of J i J = η() t P η(), () where P i the olution of (6). Suppoe that all the zero are nonminimum phae. Then A i noningular and the initial condition for the zerodynamic ubytem i η() = A B r. Uing thi value in (), and auming r =, we obtain m J =2traceA =2. (2) q i i= where q i, i =,...,m, are the nonminimum phae zero of the ytem. The above reult, obtained by Qiu and Davion (993), ay that the mallet achievable L 2 norm for the tracking error (i.e., in the limit a ε ) i larger the cloer the nonminimum phae zero are to the imaginary axi. We end thi ection with a review of the aymptotic input-output propertie of the cheap tracking controller (Kwakernaak and Sivan, 972). Let the tranfer function of the ytem () be m i= G() =α ( q i) n i= ( p, α. (3) i) The tracking controller that achieve zero teady tate error i u(t) = K εx(t)+h c ()r, (4) where K ε = ε 2 Bt P (ε) i the optimal feedback gain correponding to the olution of (4), and H c() =C(I (A BK ε)) B. Uing (4) in cloed loop with the ytem (), and taking ε, the cloed-loop tranfer function T () fromthe reference R() to the output Y () approache m + ( ) q T () i α 2 2(n m) χ n m ( ), ω =, ω + ε i= q i (5) where χ n m i a Butterworth polynomial of order n m and radiu, ω i the aymptotic radiu of the Butterworth configuration of the n m cloed loop pole that tend to infinity, and q i, i =,...,m, are the zero of the open loop tranfer function (3), which are aumed to be non-minimum phae. In (5), we can ee the two-time cale behavior of the cloed loop ytem previouly mentioned. 3. CHEAP CONTROL PERFORMANCE LIMITATIONS OF INPUT CONSTRAINED LINEAR SYSTEMS We next proceed to the cae of performance limitation when the input i contrained. The fundamentally important roll played by input contraint (both amplitude and rate) ha been eloquently explained elewhere ee for example (Stein, 989). Our aim here i to give additional theoretical upport to the undertanding of thi iue. We propoe a a meaure of the performance limitation in the preence of contraint on the magnitude of the control the value of the L 2 norm of the tracking error when a unit tep output reference i applied to a aturated limiting optimal compenator. Specifically, we propoe a a meaure of performance for open-loop table ytem of the form () ubject to the contraint u(t) t, the value of the cot function defined a J lim e 2 (t)+ε 2 u(t) 2 (t)dt, (6) ε when a unit tep reference, i.e., r(t) = (t), i applied with the ytem initially at ret, uing the control law u(t) =at ( Kx(t)+Hc ()r(t)), where if z>, at (z) z if z, if z<. and K = lim K ε. (7) ε To evaluate the cot we aume that input aturation occur in the firt part of the evolution of the ytem: Aumption. (A.) The control of the cloedloop ytem witche between the aturation level during a period of time [ +,t at ) and thereafter it never reache the aturation level again. Note that thi i a reaonable aumption given the high gain nature of the cheap controller. Under aumption (A.), the cot(6) can be eparated into two component; one correponding to the period of time where the control aturate and the other correponding to the period of time tarting when the control leave aturation until it reache the final teady tate value u( ), i.e., tat J = lim e 2 (t)+ε 2 u 2 (t)dt+ lim e 2 (t)+ε 2 u 2 (t) ε ε t at (8) We next evaluate each of the two component. Cot during aturation In order to evaluate the firt term of the cot (8), we hall ue the aymptotic propertie of the cheap tracking controller, pecifically the cloedloop tranfer function (5), to find an inputoutput equivalent open-loop controller. We ee that in order to obtain (5) by an openloop equivalent controller Q(), it mut atify G()Q() = T (), that i, thi controller hould eentially approximate the invere of the ytem. Hence, the open-loop controller for the ytem (3) equivalent to the cheap controller (4) (i.e., the one that achieve the ame cloed-loop tranfer function (5)) ha the form Q() =G MP ()F (), F() = χ n m ( ω ), (9)

where χ n m i a in (5), and G MP () ha the ame pole of G() and the reflection through the imaginary axi of the non-minimum phae zero of G() (recall that to highlight the methodology, we have aumed that all the zero of G() are non-minimum phae). Uing the equivalent open-loop controller, the problem of the aturated limiting optimal compenator can be poed uing the cheme hown in figure, i.e., thi cheme reproduce the control u = at ( Kx(t)+Hc ()r(t)). In thi cheme, Q() r(t) û(t) u(t) y(t) q at ( ) G() Q() q Fig.. Anti-windup open-loop control. i the bi-proper tranfer function (9), and q i it high frequency gain. It i eay to how that when the ytem i not aturated, the loop in figure reduce to Q(). From figure, we ee that after applying a unit tep ignal in r(t), the control u(t) typically aturate ince the gain q i uually large for mall value of ε. We thu aume, without lo of generality, that u( + ) =. The control will then witch between and until t = t at when it leave aturation to continue with a linear evolution. The crucial tep in thi analyi i then to evaluate the witching time during the aturated regime. For clarity of expoition, we illutrate the idea by taking the cae of firt and econd order ytem with, at mot, one right half plane zero. For thee cae, we have only one witch in the aturation regime. In thi cae, during the aturation period [ +,t at ), û(t) (i.e., the ignal at the input of the aturation function, ee figure ) i given by û(t) =q ( L {(Q() q ) }), (2) where L { } denote the invere Laplace tranform operator. In addition, the control ignal leave aturation when the following condition i atified û(t at) =. (2) Uing (2) and (2) we can determine the time intant t at at which the control leave aturation. Alo from figure, we ee that the tracking error during the aturation interval can be calculated a e(t) =L {G() }. With the expreion for e(t) andt at, all the ingredient to evaluate the firt term of (8) are available. After performing the limit and the integration, we can obtain an analytical expreion for the firt term of the cot. Cot after aturation Once the control ignal leave aturation, the problem reduce to the uncontrained cheap control compenator. To find the aociated cot we will ue the propertie of the low evolution of the cheap control tate in the ingular hyperplane. The firt tep i to recognize that when the ytem leave aturation, the tate i on the ingular hyperplane. Thi i eay to how by contradiction: Suppoe that the ytem leave aturation and never aturate again, and alo that the tate i not on the ingular hyperplane. Then a the control i not aturated the ytem behave like the uncontrained problem, and ince the tate i not on the ingular hyperplane there will be a ingularity in the control that will make the control aturate. Therefore, once the control leave aturation, the tate mut be on the ingular hyperplane. Conequently, once the control leave aturation at t = t at, the tate of the ytem follow the ame trajectorie that the uncontrained tate would have followed if it had tarted from the initial condition [ η(t at ) z(t at )]. Hence, the cot after aturation i e 2 (t)dt = η(t at) P η(t at), (22) t at where η(t at ) i determined from [ ] [ ] tat η(t at) = e Ant η() at + e An(t τ) B ndτ, z(t at) z() (23) where A n [ ] A [B ] A A 2 [ B n ]. B Equation (22) allow u to complete the calculation of the cot by evaluating the econd term of of the right hand ide of (8). For ytem with only one non-minimum phae zero, the evaluation of the cot once the ytem leave aturation can be coniderably implified ince it i not neceary to olve equation (23). The procedure i explained in the following. If the ytem ha only one non-minimum phae zero the ingular hyperplane become a line in R n. Indeed from () and (3) we ee that η =(A B B t P ) η, z = B t P η, z 2 = B t P (A B B t P ) η, z 3 = B t P (A B B t P ) 2 η,. z n = B t P (A B B t P ) n 2 η. Once the ytem leave aturation, it behave like the uncontrained cheap control problem, and ince the ingular hypeplane i a line, the output e = z after aturation ha the ame evolution a the uncontrained output hifted in time. A a conequence, we can evaluate the cot after aturation of the contrained cheap control problem uing a partial cot of the uncontrained cheap control problem a

2.8.6.4.2.8.6.4.2.2.2.4.6.8 J.5. τ = η..2.4.6.8 Singular Hyperplane Saturated Cheap Control Trajectory.5 τ =. τ =. 2 4 6 8 2 4 6.2 Fig. 2. Phae portrait for uncontrained and aturated cheap tracking control. [( ( L q + ) q + z ] ) 2 dt = 2 q e 2qt. e 2 dt t at t (24) The approximation in (24) come from neglecting the high frequency pole in the Butterworth arrange of (5). The value t i determined from the condition [ L G() ] = L t at [( q + q + ) ] Comparing (24) with (2) for the cae of a ingle non-minimum phae zero, we ee that the partial cot (24) i maller than the total uncontrained cot (2). However, wherea the tranition to the ingular hyperplane i cotle in the uncontrained cae, it ha a nonzero cot in the contrained cae ince the tate cannot jump to the ingular hyperplane but ha a low evolution while the control i aturated. The combination of the two partial cot yield a cot larger than the uncontrained cot (2), a we will ee in the example of ection 4. To illutrate the above idea, figure 2 how a imulation reult for the econd order example explained in detail in ection 4. In thi figure, we ee the tate trajectory uing cheme of figure, and alo uing uncontrained cheap control. We next preent two example. 4. EXAMPLES Example : Conider the following ytem G() = τ+. Then, the equivalent open loop cheap controller i given by Q() = τ+ β +, where β = ɛ. It alo follow that, q = τ β Q() q (τ β) = τ(τ+) Uing (25) and (26) in (2) we have û(t) = τ [ ] (τ β) ( e τ t ) β τ t. (25) (26) Fig. 3. Cot v. for example Thi lat expreion i valid until t at which, uing(2), and taking the limit a β i found to be [ ] t at τln (27) On the other hand, until t at the tracking error i e(t) = ( e τ t ) (28) Finally, ince the ytem i minimum phae, and uing (28) and (27), we obtain the following expreion for the cot (8): [ ] J = τ( ) 2 ln ( [ 2( ) + τ ] τ 2 ) 2 2τ ( [ ] 2τ 2 (29) Figure 3 how a graphical repreentation of the value of the cot (29) a a function of for different value of τ. It hould be noted that the et point value of the control ignal hould be feaible, i.e., > for thi cae. Note that ince thi ytem i minimum phae, the limiting cheap cot for i zero. Alo, a expected, the fater the plant, then the lower the cot, ince low plant require more control effort that contribute to aturation in thi cae. However, the relation (29) i not trivial. Example 2 : Conider the following ytem G() = 2( σ) ( +)( +2) In thi cae, the equivalent open loop cheap controller i given by Q() = It then follow that: Q() q = ( +)( +2) 2( + σ)( + β). q = 2σβ 2[(σ + β 3σβ) +( 2σβ)] ( +)( +2) (3) (3) Uing (3) and (3) in (2) we have that a β t at ln σ (σ ) 2 +(+2σ) ( ) 2σ (32) The tracking error on the interval [ + t at )i e(t) = [ 2( + σ)e t +( 2σ)e 2t] (33) )

J.2.8.6.4 σ =.4 σ =.2 σ =..2 5 5 2 25 3 35 4 45 5 Fig. 4. Cot v. for example 2 The value t i given by [ t ( + σ) = σ ln + e t at 2 2 ] ( + 2σ) e 2t at 2 (34) Uing (32), (33), (34) and (24) we can evaluate the cot numerically. The reult are hown in figure 4 for three different poition of the non minimum phae zero and plot the cot veru. Note that a the limiting cot approache 2σ, which i conitent with the reult of uncontrained cheap control. The reult hown in figure 4 give inight into the effect of the input contrained achievable performance. It i intereting, for intance, to note that a contraint = 5 (which i five time the teady tate input neceary in thi cae) change the performance limit aociated with a non minimum phae zero at (σ =.) to be equivalent to the performance limit achieved without contraint for a non minimum phae zero at 5. Thi illutrate the fact that, depending on condition, the effect of input contraint can wap linear effect due to right half plane pole or zero. Thi i in accord with intuition. 5. CONCLUSIONS Under the aumption that the ytem witche between the aturation level only for a certain period of time and never thereafter, the cot can be evaluated in term of two component. The firt component correpond to the aturation period. We preent a way to evaluate thi component and non trivial analytical expreion are obtained a a function of the contraint level. The econd term correpond to the period of time from the intant the control leave aturation until the control reache it teady tate. Thi cot i zero for minimum phae ytem and different from zero for non minimum phae ytem. For the non minimum phae cae, we give a way to calculate thi cot, which i very imple if the ytem ha only one non minimum phae zero. The propoed method eem to be of value in gaining a deeper undertanding of the relationhip between the dynamic of a ytem and the contraint eential at the tage of determining the authority of the actuator for ome application. 6. REFERENCES Bode, H. (945). Network Analyi and Feedback Amplifier Deign. D. van Notrand, New York. Franci, Bruce A. (979). The optimal linearquadratic time-invariant regulator with cheap control. IEEE Tran. on Automat. Contr 24(4), 66 62. Iidori, A. (995). Nonlinear Control Sytem. 3rd. ed.. Springer-Verlag. Kokotović, P., H. Khalil and J. O reilly (986). Singular perturbation methith in control: Analyi and Deign. Academic Pre. Kwakernaak, H. and R. Sivan (972). Linear Optimal Control ytem. Wiley. Middleton, R. H. and J. H. Bralavky (2). On the relationhip between enitivity integral and limiting optimal control problem. Proceeding of Conference on Deciion and Control. Sydney, Autralia. Qiu, L. and E.J. Davion (993). Performance limitation of non-minimum phae ytem in the ervomechanim problem.. Automatica 29(2), 337 349. Saberi, A. and P. Sannuti (987). Cheap and ingular control for linear quadratic regulator. IEEE Tran. on Automat. Contr 32(3), 28 29. Serón, M. M., J. H. Bralavky and G. C. Goodwin (997). Fundamental Limitation in Filtering and Control. Springer Verlag. Serón, M.M, J. H. Bralavky, P.V. Kokotović and D.Q. Mayne (999). Feedback limitation in nonlinear ytem: From bode integral to cheap control. IEEE Tran. on Automat. Contr 44(4), 829 832. Stein, G. (989). Repect the untable. IEEE Control ytem Society Bode Lecture.