Neste Saturation with Guarantee Real Poles Eric N Johnson 2 an Suresh K Kannan 3 School of Aerospace Engineering Georgia Institute of Technology, Atlanta, GA 3332 Abstract The global stabilization of asymptotically nullcontrollable linear systems with boune controls has been stuie extensively An early contribution was by Teel [6] who propose a set of neste saturators to globally asymptotically stabilize the special case of n- integrators with one input Using this law however, the close loop system pole locations epen on the choice of coorinate transformation use to arrive at the control law In this paper we suggest an approach that allows the esigner to pick transformations that facilitate the placement of the close loop poles on the negative real axis Introuction The problem aresse involves the global stabilization of a chain of integrators ẋ = x 2,, ẋ n = u () The system given by () is a subset of a class of systems that are sai to be asymptotically null-controllable with boune controls [3, ] This property was shown in [2] to be equivalent to the system being stabilizable an having all open-loop poles in the close left-half plane It was shown in [4] that it is not possible to globally stabilize integrator chains of orer n > 2 using a boune linear feeback law However, it was shown by Teel in [6] that a nonlinear law consisting of neste saturators can guarantee global asymptotic stability for integrator chains of any orer n This control law may be expresse as u = σ n (h n (x) + σ n (h n (x) + + σ (h (x)))) where h i are linear combinations of the state (feeback) an the saturation functions σ i satisfy certain properties The existence of such a globally stabilizing control law was establishe in [6] by choosing one set of h i s This work was supporte in part by DARPA contract #3365-98-C-34 2 Lockhee Martin Assistant Professor of Avionics Integration ericjohnson@aegatecheu 3 Grauate Research Assistant suresh kannan@aegatecheu -783-7896-2/3/$7 23 IEEE 497 such that global asymptotic stability coul be proven The choice of h i is a esign egree of freeom an may be exercise to prescribe pole locations an the linear ynamics when ifferent elements of the control law are saturate We observe that the h i chosen by Teel with conventional saturation functions (see Definition 2) results in all the poles of the close loop system resiing at when none of the saturation elements in the control law are saturate If the k th saturator is the outermost element to be saturate, then the resulting close loop system has poles at with multiplicity n k an poles at with multiplicity k, at least until the element comes out of saturation A iscussion on the prescription of performance by pole placement (both real an complex) is provie in [5], however no explicit transformation is provie Another aspect is the behavior of these poles as ifferent elements of the control law saturate Ieally, these poles shoul not change when saturation occurs Both these properties (pole placement an movement when saturate) are useful if the neste saturation control law is to be employe in practice We believe that the simple an elegant neste saturation law can benefit greatly from these properties Hence, the effort here is to evelop a transformation, ie, a way to select h i such that close loop poles for the unsaturate system may be prescribe as { a, a 2, a n }, where a i R \ an a i > for stability Aitionally, it will be shown that when the outermost saturate element is σ k, the poles of resulting linear system resie at { a, a 2, a n k,, 2, k } 2 Main Result Definition (Linear saturation) Define constants (L, M) R+ such that < L M Now, efine a function σ : R R σ is sai to be a linear saturation if it is continuous, nonecreasing an satisfies a sσ(s) > s b σ(s) = s when s L c σ(s) M s R Denver, Colorao June 4-6, 23
Definition 2 (Conventional saturation) σ : R R is sai to be a conventional saturation if it has a limit M R+ such that a sσ(s) > s b σ(s) = s when s M c σ(s) = M when s > M Remark σ is sai to be saturate when its argument is not in its linear region For linear saturation this occurs when s > L For conventional saturation this occurs when s > M Remark 2 Conventional saturation is a special case of linear saturation with L = M an a constant saturation value M Lemma Consier a chain of n-integrators, given by (), which may be represente as ẋ = A x x + B x u, with x R n, u R an A x =, B x = (2) then there exists a linear transformation y = T yx x which transforms () into ẏ = A y y + B y u where, a n a n a n a n A y =, B y = a 2 an the elements a i R \ with i = n Proof: Given a set of coefficients a n a n a (3) A = {a, a 2,, a n } (4) Let A l A represent a subset containing the first l elements of A Define a function Fk m(a l) which acts over the set A l Fk m is use to generate the prouct of combinations of elements taken m at a time from A l The number of ( such) combinations is given by the binomial l coefficient Hence, F m k m(a l) may be treate as a generating function that outputs the k th combination of the prouct of m elements taken from the set A l without repetition an isregaring orer Note that Fk = 498 In orer to generate the transformation T yx, efine the function C(l, m), with l [,, n], m [,, l] an m l, over the set of coefficients A given by (4) C(l, m) = C l m k= F m k (A l ) (5) C(l, ) = (6) ( l C m l is the binomial coefficient The new coori- m) nate system is characterize by y n i = a i+ i C(i, j)x n j, i [,, n ] an the transformation T yx is explicitly given by T yx(n i)(n j) = a i+ C(i, j) i j (7) T yx(n i)(n j) = i < j (8) for i, j [,, n ] Aitionally, T yx is an upper iagonal matrix with non-zero iagonal entries Hence, T xy = Tyx exists Finally, observing that ẋ = T yx A x T yx y + T yx B x u = A y y + B y u it is enough to verify that A y T yx = T yx A x an that T yx B x = B y This may be carrie out using Equations 2, 3 an 8 Theorem For the system given by () Given any set of positive constants {(L i, M i )}, where L i M i for i =,, n an M i < 2 L i+ for i =,, n, an for any set of functions {σ i } that are linear saturations for {(L i, M i )}, there exists a linear coorinate transformation y = T yx x such that the boune control u = σ n (y n + σ n (y n + + σ (y ))) (9) results in a globally asymptotically stable system Proof: In short, use the transformation given by Lemma in the proof of Theorem 2 in [6] It is however restate here for completeness Use the coorinate transformation y = T yx x given by Lemma an choose the set of coefficients a i > Substituting the neste saturation law given by Eq (9) into Eq () an expaning yiels the close loop system ẏ = a n [y 2 + +y n σ n (y n + σ n ( σ (y )))] ẏ 2 = a n [y 3 + +y n σ n (y n + σ n ( σ (y )))] () ẏ n = a 2 [ y n σ n (y n + σ n ( σ (y )))] ẏ n = a σ n (y n + σ n ( σ (y ))) Denver, Colorao June 4-6, 23
The trajectory of y n is examine first Choosing a Lyapunov function V n = y 2 n, with y n R Its erivative V n may be written as V n = 2a y n [σ n (y n + σ n (y n + + σ (y )))] Noting that a i > Definition, conitions (a), (b), imply that y n an σ n ( ) are the same sign only if y n + σ n ( ) is the same sign as y n Conition (c) of Definition applie to σ n an having chosen M n < 2 L n, it can be seen that V n < for all y n / Q n = {y n : y n 2 L n} If starting outsie Q n, the trajectory of y n eventually enters Q n in finite time Since the RHS of Eq () is globally Lipschitz, the erivatives are boune resulting in the remaining states y y n remaining boune for any given finite time Once y n has entere Q n, conition (b) of Definition implies σ n operates in its linear region because the argument to σ n is boune as y n + σ n ( ) 2 L n + M n L n The equation for the evolution of y n is now given by ẏ n = a 2 y n a 2 y n a 2 σ n (y n + + σ (y )) = a 2 σ n (y n + + σ (y )) which is similar to the expression for ẏ n Using similar arguments as that use for the evolution of y n, it can be shown that y n enters a set Q n in finite time an remains in Q n thereafter with all remaining states being boune Continuing in the same fashion, it can be shown that every state y i for i [,, n], enters a set Q i = {y i : y i 2 L i} in finite time an all saturation functions σ i are operating in their linear regions Hence after a certain finite amount of time the governing equations, Eq (), becomes ẏ = a n y ẏ 2 = a n (y + y 2 ) ẏ n = a (y + y 2 + + y n ) which is exponentially stable Corollary (Pole location) If the saturators use are Conventional saturation, an none of the σ i are saturate, the poles of the linearize close loop system resie at { a, a 2,, a n } During perios when the outermost saturate element is the k th saturator, σ k, the poles of the resulting close loop linear system resie at { a, a 2,, a n k,, 2,, k } 499 Proof: Using the neste saturation law, the closeloop n-integrator system may be expresse as ẋ n + σ n (y n + σ n (y n + + σ (y ))) = When the k th saturator is saturate, an σ k+ σ n are not saturate, the close loop system is given by ẋ n + y n + y n + y k+ ± M k = This represents a force linear system where the forcing function is the constant M k Examining the homogeneous part = ẋ n + y n + y n + + y k+ Using Eq (7) to expan y i = ẋ n + a C(, j)x n j + a 2 C(, j)x n j + n (k+) + a n k C (n (k + ), j) x n j Noting that x = x, ẋ = x 2,, x (n ) = x n, x (n) = ẋ n, an substituting p = n k for clarity the characteristic equation may be written as Υ(λ) = λ n + a C(, )λ n + a 2 C(, )λ n + a 2 C(, )λ n 2 + a p C(p, )λ n + + a p C(p, p )λ k Factoring out λ k Υ(λ) =λ k [λ p + a C(, )λ p + a 2 C(, )λ p + a 2 C(, )λ p 2 + a p C(p, )λ p + + a p C(p, p )] an may be written in its final form as Υ(λ) = λ k (λ + a )(λ + a 2 )(λ + a p ) which has k zeros an p = n k non-zero stable poles at known locations Corollary 2 During perios when σ k is the outermost saturate element in the control law of Theorem an the coorinate transformation use is given Denver, Colorao June 4-6, 23
by Lemma, then, in steay-state, the magnitue of the k th erivative, ẋ k, is given by ẋ k = M k a n k C(n (k + ), n (k + )) () for k [,, n ] for k = n ẋ k = M k (2) Proof: If σ k is saturate, the close loop system may be written as Using Eq (7) ẋ n + y n + y n + y k+ ± M k = = ẋ n + a C(, j)x n j + a 2 C(, j)x n j + (3) n (k+) + a n k C (n (k + ), j) x n j ± M k When the outermost saturate element is σ k, the ynamics eventually reach a saturate-equilibrium region where higher-orer erivatives reach zero So, x (n) x (k+), ie, ẋ n, x n x k+2 go to zero The only term left from Eq (3) is a n k C(n (k + ), n (k + ))x k+ ± M k = (4) Noting that ẋ k = x k+, rearranging Eq (4) an taking the absolute value of both sies results in Eq () Finally, when k = n, the outermost saturator σ n is saturate an Eq (3) reuces to If x (n) (t) L n+ ɛ for all t t an for some ɛ > an given linear saturation functions σ i with parameters (L i, M i ) satisfying, L i M i i =,, n + M i < 2 L i+ i =,, n M n ɛ then, the feeback u = x (n) σ n (y n + σ n (y n + + σ (y ))) with y = T yx e given by Lemma, where, e i = x i x (i ) for i = n, results in a globally asymptotically stable system Aitionally if conventional saturation elements are use, the error ynamics are governe by Corollary an quasi-steay rates governe by Corollary 2 Proof: The ynamics of Eq (6) may be expresse in terms of the error e ė = e 2,, ė n = x (n) + σ n+ (u) With the given control law, if the magnitue of the n th erivative of the comman x is always such that x (n) (t) L n+ ɛ for all t t an M n ɛ, then the magnitue of the argument of σ n+ is x (n) σ n ( ) L n+ an σ n+ is always in its linear region, resulting in the close loop error ynamics becoming ė = e 2,, ė n = σ n (y n + σ n (y n + + σ (y ))) (8) The conitions of this corollary an form of Eq (8) satisfy the requirements of Theorem This implies that the ynamics of e are asymptotically stable an hence x tracks x asymptotically The form of Eq (8) also allows Corollary an Corollary 2 to be applie irectly ẋ n ± M n = (5) Rearranging Eq (5) an taking magnitues of both sies results in Eq (2) Corollary 3 (Restricte Tracking) Consier a nonlinear system with magnitue saturation at the input u given by ẋ = x 2,, ẋ n = σ n+ (u) (6) an a compatible reference signal given by [ x (t), ẋ (t), x (n) (t) ] (7) 5 3 Examples Global Stabilization: Consier the problem of stabilizing the 3 r orer system ẋ = x 2, ẋ 2 = x 3, ẋ 3 = u using boune control u [, ] (conventional saturation) with poles at {, 3, 2} Then, {a, a 2, a 3 } = {, 3, 2} The transformation require to achieve these poles may be expresse as y y 2 = a 3(a a 2 ) a 3 (a + a 2 ) a 3 a 2 (a ) a 2 x y 3 x 2 a x 3 Denver, Colorao June 4-6, 23
σ x 4 3 2 States 5 5 2 25 3 35 4 45 5 5 Saturation Functions 5 5 5 2 25 3 35 4 45 5 time /s Figure : Initial conition response of a 3 r orer system Using the neste saturation law given by Theorem an choosing the saturation element parameters as follows M 3 =, L 3 = M 3 M 2 = 2 L 3 ɛ, L 2 = M 2 M = 2 L 2 ɛ, L = M where ɛ is a small positive number, that is use to satisfy the inequality M i < 2 L i+ Aitionally the saturation element parameters are chosen L i = M i (Conventional saturation) Then, the close loop system is given by ẋ 3 + σ 3 (y 3 + σ 2 (y 2 + σ (y ))) = An initial conition response with x = [, 5, ] is shown in Figure The figure also shows the outputs of the saturation elements - 5s, σ 3 is saturate 5-3s, σ 2 is saturate 3-46s, σ is saturate 46-5s, control law is unsaturate The only region where the system practically reaches a saturate-equilibrium is when σ is saturate, between an 4 secons The equilibrium value for ẋ is given by Corollary 2 ẋ = x 2 = M a 2 a = 833 an matches the slope of x in Figure x x 2 x 3 σ σ 2 σ 3 5 Restricte Tracking: Consier a chain of 4 integrators where, σ 5 (u) represents a magnitue saturate actuator ẋ = x 2, ẋ 2 = x 3, ẋ 3 = x 4, ẋ 4 = σ 5 (u) σ 5 is a conventional saturation function with parameters (L 5, M 5 ) A compatible comman may be represente as [x, ẋ, ẍ, x, x ] Defining the error as, e = x x, the error erivatives may be written as ė = x 2 ẋ ë = x 3 ẍ e = x 4 x e = σ 5 (u) x The control is given by Corollary 3 u = x σ 4 (y 4 + σ 3 (y 3 + σ 2 (y 2 + σ (y )))) with x L 5 ɛ an M 4 ɛ, for some ɛ >, the saturation function parameters (L i, M i ) chosen to satisfy the conitions given by Corollary 3 an y i given by Lemma The coorinate transformation use is y = T yx e where T yx is given by a a 2 a 3 a 4 (a a 3 + a 2 a 3 + a a 2 )a 4 (a + a 2 + a 3 )a 4 a 4 a a 2 a 3 (a + a 2 )a 3 a 3 a a 2 a 2 a Here, the poles were taken to be at { a, a 2, a 3, a 4 } = { 5,, 2, 3} The saturation function parameters were chosen as L 5 = ɛ = 2 L 5 L 4 = ɛ L 3 = 2 L 4 L 2 = 2 L 3 L = 2 L 2 M 5 = L 5 ɛ M 4 = ɛ ɛ M 3 = L 3 ɛ M 2 = L 2 ɛ M = L ɛ where ɛ is a small positive number chosen to satisfy M i < L i If Corollary 2 is evaluate for various saturation elements being saturate ė 4 = M 4 when, σ 4 is saturate ė 3 = M 3 a when, σ 3 is saturate ė 2 = M 2 a a 2 when, σ 2 is saturate ė = M a a 2 a 3 when, σ is saturate (9) Denver, Colorao June 4-6, 23
The response of this system to a sinusoial comman compatible with x = 5 sin(5t) an zero initial conitions is illustrate in Figure 2 From Eq (9) notice that as the banwith ie, a i is increase, the error rates in saturate-equilibrium ecrease Hence for higher banwith, the overall settling time can be higher, which is perhaps counter-intuitive This aspect is further illustrate in Figure 3 where it is observe that the control law with faster poles (all at -5) takes longer to be regulate back to than the system with slower poles (all at -5) The initial conition use was x = [,,, 2] T with zero comman x, x 5 5 5 5 2 25 3 35 4 45 5 x vs x Saturation Functions x x 5 σ σ 2 4 Conclusion The extensions presente thus far provies a transformation that allows placement of poles on the non-zero real axis Assuming the poles chosen are stable, global asymptotic stability is guarantee These chosen poles are guarantee to remain constant, apart from going to zero when the respective saturation element saturates Finally, when in saturate-equilibrium ue to σ k saturating, the quasi-steay rate of state change is given by a Corollary σ 5 5 5 5 2 25 3 35 4 45 5 time /s Figure 2: Response to a sinusoial comman for a 4 th orer system σ 3 σ 4 References [] Tingshu Hu an Zongli Lin Control Systems with Actuator Saturation : Analysis an Design Birkhaüser, 2 [2] Euaro D Sontag An algebraic approach to boune controllability of linear systems 39:8 88, 984 [3] Hector J Sussmann, Euaro D Sontag, an Yui Yang A general result on the stabilization of linear systems using boune controls IEEE Transactions on Automatic Control, 39(2), ec 994 [4] Hector J Sussmann an Yui Yang On the stabilizability of multiple integrators by means of boune feeback controls In Proceeings of the 3th IEEE Conference on Decision an Control, Brighton, UK, 99 [5] Anrew R Teel Feeback stabilization : Nonlinear solutions to inherently nonlinear problems Technical Report UCB/ERL M92/65, Electronics Research Laboratory, University of California, Berkeley, CA 9472, June 992 [6] Anrew R Teel Global stabilization an restricte tracking for multiple integrators with boune controls Systems & Control Letters, 8:65 7, 992 x 4 2 8 6 4 2 x reponse for fast an slow poles Poles at 5 Poles at 5 2 3 4 5 6 7 8 time /s Figure 3: Comparison of the initial conition response, for a 4 th orer system The soli curve settles faster an has all poles at -5 whilst the ashe-curve settles slower an has poles at - 5 52 Denver, Colorao June 4-6, 23