The Real Stabilizability Radius of the Multi-Link Inverted Pendulum

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1 Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, Jne 14-16, 26 WeC123 The Real Stabilizability Radis of the Mlti-Link Inerted Pendlm Simon Lam and Edward J Daison Abstract A mlti-link inerted pendlm with an arbitrary nmber of links and controlled by a single torqe inpt is considered in this paper It is well known that as the nmber of pendlm links increases, an experimental pendlm system becomes more difficlt to stabilize, and this is demonstrated in this paper for a nonlinear model of a mlti-link inerted pendlm system It is conjectred that the primary reason for sch an increase of difficlty is de to the poor stabilizability robstness properties of the pendlm s linearized model as additional links are added Using the real stability and stabilizability radis, this conjectre is confirmed I INTRODUCTION In the literatre, a good deal of sccess has been reported for stabilizing an experimental single-link and doble-link inerted pendlm system, while there are only a few reported reslts for the triple pendlm (eg 1) For stabilizing pendlm systems with for or more links, no experimental sccess can be fond in the literatre Ths it appears that as the nmber of links increases, an experimental inerted pendlm system becomes more challenging to balance, and this begs the qestion of why ie why does an inerted pendlm system become more difficlt to stabilize as the nmber of links increases? Generally, ery few stdies hae been carried ot related to mlti-link inerted pendlm systems Among papers which do so, they mainly deal with deriing the nonlinear dynamics (eg 2, 3) and determining the controllability of the system (eg 4) In terms of answering the qestion, some possible factors hae been sggested In 5, it is claimed that a mlti-link inerted pendlm is ery sensitie to torqe distrbances, joint friction, and measrement noise In 6, other difficlties are reported that inclde system complexity reslting from filters sed to redce measrement noise, system ncertainties de to friction effects, and controller satration de to high gains reslting from trying to control the links Instead of stdying sch indiidal factors, the main reslt of this paper is the stdy of the oerall stabilizability robstness properties of a linearized mlti-link inerted pendlm system In particlar, it is shown that a fndamental difficlty is that the system s real stabilizability radis becomes arbitrarily small as the nmber of links increases To start, the nonlinear eqations and state-space model of a mlti-link pendlm system is obtained in Section II, and the linearized system model is proed to be controllable and obserable in Section III The nonlinear (NL) pendlm This work has been spported by NSERC nder grand No A4396 S Lam and EJ Daison is with the Systems Control Grop, Department of Electrical and Compter Engineering, Uniersity of Toronto, Toronto, ON, Canada, M5S 3G4 simon,ted@controltorontoca Fig 1 θ 1 θ 2 m 1 m 2 θ l 2 Model of a mlti-link inerted pendlm with links system is then shown in Section IV to indeed be progressiely more difficlt to stabilize as the nmber of links in the system increases In Section V, the real stabilizability radis of the system is obtained, which is shown to become excessiely small as the nmber of links increases, which implies that there exists no practical controller which can stabilize the linearized system with a large nmber of links Finally, the same mlti-link pendlm system with mltiple inpts is then stdied l m II MODELING THE PENDULUM SYSTEM A Nonlinear Dynamics Fig 1 illstrates a mlti-link inerted pendlm system with links, where the i-th link is modeled as a pointmass, m i, attached ia a massless rigid rod of length, l i, for i = 1,, The control inpt,, is a single torqe applied at the piot of the bottom link All angles are measred with respect to the ertical Defining x = T θ1 θ θ1 θ as the state ector, and on ignoring Colomb and iscos friction effects, the anglar accelerations of the links can be written as: θ 1 G(x) = F(x,) (1) θ where the elements of the matrices G and F are: G ij = l j cos (θ i θ j ) (2) m k k=max(i,j) /6/$2 26 IEEE 1814

2 F i = m h l k sin (θ k θ i ) θ k 2 k=1 h=max(k,i) ( ) + m k g sin θ i + δ i1 1 k=i {, i j for i, j =1, 2,,, δ ij,andg =98 1, i = j m s 2 Here the nits of the masses and lengths are assmed to be in kg and m respectiely B Linearized State-Space Model Linearizing (1) abot the zero eqilibrim point, (x,)= (, ) (and assming that only the bottom link s angle is otptted), we obtain the following state-space representation: ẋ = Ax + B, y = Cx (4) where A = (M L ) 1, B = M a (M L ) 1, C = 1, where M b ( ) M a = diag g m i g m i m g, i=1 i=2 1 m 1 m 2 m M b =, M m 2 m =, m l 2 and L = l 2 l III CONTROLLABILITY &OBSERVABILITY The following reslts are obtained; whose proof may be fond in 7: Theorem 31: The -link inerted pendlm s linearized system (4) is controllable for all 1 Corollary 31: The -link inerted pendlm s linearized system (4) is obserable for all 1 IV DIFFICULTY OF BALANCING A NL MULTI-LINK PENDULUM SYSTEM MODEL Since the -link inerted pendlm system is controllable (and obserable) for any 1, this implies from linear control theory that for a pendlm system with any nmber of links, a linear feedback controller is garanteed to exist that can locally stabilize it Howeer, de to real-life factors, this is not experimentally tre, especially for a pendlm system with a large nmber of links Three of these real-life factors are looked at in this section: (i) initial conditions; (ii) external physical distrbances; and (iii) parametric ncertainties In the following nonlinear simlation examples, the nominal -link pendlm system is assmed to hae nit link masses and lengths (ie m i = l i =1for i =1, 2,,) (3) TABLE I MAXIMUM STABLE INITIAL ANGLES (IN DEGREES) Initial Configrations Aligned Bottom-only Alternating The nominal controller applied is an optimal state-feedback controller = kx (5) obtained by soling the Linear Qadratic Reglator (LQR) problem and minimizing the following performance index: J = (y Qy + ɛ ) dτ (6) where Q =1and ɛ =1, for the linearized system (4) A Simlation #1: Initial Conditions Since the dynamics of the pendlm system is highly nonlinear, the linear controller (5) can only stabilize the system locally abot the ertical So in this simlation, it is desired to find the maximm initial angles (with zero anglar elocities) sch that the nonlinear closed-loop system is stable, sing the controller (5) θ θ θ θ +θ +θ (i) (ii) (iii) Fig 2 Varios initial configrations: (i) aligned; (ii) bottom-only; and (iii) alternating Howeer, instead of performing a -D grid search for the maximm stable initial angles, we choose three initial configrations (see Fig 2), where each is parametrized by a single ariable, θ, and a 1-D search is performed to obtain the maximm ale (θ max ) sch that the corresponding nonlinear pendlm system is stable for all θ θ max,θ max In this simlation, the system behaior is considered to be nstable if any of the link angles become larger than 9 The reslts obtained are gien in Table I for = 1, 2,,7 For each of the three initial configrations, it can be seen that as increases, the range of stable initial angles decreases, which illstrates the difficlty in balancing a pendlm system as the nmber of links increases θ 1815

3 B Simlation #2: External Physical Distrbances In the second simlation, it is desired to examine how mch distrbance the nonlinear closed-loop system can tolerate before the controller (5) fails to maintain BIBO stability In particlar, a white noise random distrbance bonded by b is applied to the top link (see Fig 3) +b singlar ale of ( ), andwhereδ LM is gien by: Δ Δl Δm 1 =Δ LM l m 1 Δm m (8) w b t The reslts obtained are shown in Table III It can be seen that the accracy of the system parameters needs to be increased as the nmber of links in the system increases TABLE III ALLOWABLE PARAMETRIC UNCERTAINTIES Fig 3 A bonded random distrbance is applied to the top-most link Table II gies the maximm magnitde of the distrbances, b max,thatthe-link inerted pendlm can tolerate before the system becomes nstable It can be seen that as the nmber of pendlm links increases, the system becomes less tolerant and more sensitie to external distrbances TABLE II MAXIMUM TOLERABLE DISTURBANCE b max C Simlation #3: Parametric Uncertainties A controller, designed for a particlar system with a gien set of parameters, may fail to work if the parameters change oer time, or are simply not accrate enogh Hence, the prpose of the third simlation is to inestigate how ncertainties in the parameters of the system affect the closed-loop stability of the system, as the nmber of links increases In particlar, it is desired to find the maximm allowable ncertainties in the masses and lengths of the system, sch that closed-loop stability can be garanteed For this setp, we assme that the lengths and masses of the nominal system s links are pertrbed as follows: li m i l i +Δl i m i +Δm i (7) for i =1, 2,,, and find the largest pertrbations sch that the pertrbed system remains stable sing the nominal controller (5) The (normalized) size of the pertrbations is defined as r LM = σ(δ LM ),where σ( ) denotes the largest Parameter Accracy r LM (Sig Fig Reqired) Remark 41: In the aboe three simlations, the weights of the cost fnction (6) are chosen to be Q =1and ɛ =1 Althogh it is well known that the closed-loop response of the system depends on both weights, it is erified that the response in or case depends mainly on ɛ, and that choosing ɛ =1proides the better stability performance V MAIN RESULTS In this paper, it is conjectred that the fndamental difficlty of experimentally balancing a pendlm system with many links is the lack of stabilizability robstness, as measred by the real stabilizability radis 8, of the linearized model (4) of the pendlm system This lack of stabilizability robstness of model (4), shows p as a lack of stability robstness, as measred by the real stability radis 9, of the closed-loop system (4)+(5) This conjectre is erified in the following examples of a pendlm system consisting of links with nit masses and nit lengths A Strctre of Plant and Controller Pertrbations Since the pper half of the pendlm system s (A, B) matrices (4) is fixed by definition, we consider only plant pertrbations, (Δ A, Δ B ), with the strctre: ( ) (Δ A, Δ B )=, (9) where denotes a sb-matrix where elements may be arbitrary bt real Regarding controller pertrbations, Δ k, since it is possible for any of the feedback gains to ary, we impose no strctre; ie: Δ k = (1) 1816

4 B Real Stability Radis of Pendlm 1) Absolte Real Stability Radis of Pendlm: Assme now that the system (4) is sbjected to the absolte plant pertrbations with the strctre gien by (9), ie A A + Δ A B B + Δ B (11) and that the controller (5) is applied The pertrbed closedloop system is ths described by: ẋ =(A + BΔ s,p C) x (12) I where A = A + Bk, B =, C =,and k Δ s,p = Δ A Δ B The absolte real stability radis of the pendlm, denoted as r s,p (absolte), is hence eqal to the real stability radis of the system (12) 2) Normalized Real Stability Radis of Pendlm: Now consider the following (strctred) plant pertrbations normalized by the plant s system matrices, (A, B): A A + Δ A A B B + Δ B B (13) The pertrbed closed-loop system is: ẋ = ( Ā + B Δ s,p C) x (14) A where Ā = A + Bk, B =, C =,and Bk Δ s,p = Δ A Δ B The normalized real stability radis of the pendlm, denoted as r s,p (normalized), is hence eqal to the real stability radis of the system (14) 3) Reslts: Table IV shows the absolte and normalized real stability radis of a -link inerted pendlm for = 1, 2,,7 It can be seen that the normalized real stability radis of the pendlm system qickly becomes smaller as the nmber of links increases Ths the closed-loop system loses robstness to plant pertrbations as the nmber of links increase, which implies that a higher accracy is needed in the plant parameters in order to garantee closed-loop stability The 7-link pendlm system, for example, reqires that the plant data to be accrate to 6-7 significant figres, while a 1-link pendlm only reqires 2-3 significant figres C Real Stability Radis of Controller 1) Absolte Real Stability Radis of Controller: Consider the following absolte pertrbations to the controller (5): k k + Δ k (15) The absolte real stability radis of the controller, denoted as r s,k (absolte), is eqal to the real stability radis of: ẋ =(A + BΔ s,k C) x (16) where A = A + Bk, B = B, C = I, andδ s,k = Δ k TABLE IV REAL STABILITY RADIUS OF PENDULUM r s,p(absolte) r s,p(normalized) Sig Fig Reqired ) Normalized Real Stability Radis of Controller: Assme the following normalized controller pertrbation: k k + k Δ k (17) The normalized real stability radis of the controller, denoted as r s,k (normalized), is hence eqal to the real stability radis of the pertrbed closed-loop system: ẋ = ( Ā + B Δ s,k C) x (18) where Ā = A + Bk, B = Bk, C = I, and Δ s,k = Δ k 3) Reslts: Table V shows the absolte and normalized real stability radis of the controller (5) applied to the linearized model (4) for the -link inerted pendlm system It can be seen that the real stability radis of the controller decreases dramatically as the nmber of links increases; ths, again, signifying the closed-loop system s lost of robstness to parametric pertrbations TABLE V REAL STABILITY RADIUS OF CONTROLLER r s,k (absolte) r s,k (normalized) Sig Fig Reqired D Real Stabilizability Radis of Pendlm The reslts of the absolte and normalized real stabilizability radis are now presented in this section 1) Absolte Real Stabilizability Radis of Pendlm: Gien the system (4), consider the strctred absolte pertrbations (11) The pertrbed open-loop system becomes: ẋ =(A + EΔ A F) x +(B + EΔ B G) (19) where E =, F = I, and G = 1 Howeer, from Definition A1 of the strctred real stabilizability radis, E, F, and G mst be non-singlar Hence, we ɛi approximate E by Ẽ = and let ɛ Inthis case, the absolte real stabilizability radis of the pendlm, 1817

5 r c,p (absolte), is eqal to ) the strctred real stabilizability radis of (A, B, Ẽ, F, G Note: for the prpose of implementation, ɛ is chosen as small as possible (eg ɛ =1 6 ) sch that Ẽ is inertible and not ill-conditioned 2) Normalized Real Stabilizability Radis of Pendlm: Now consider the system (19) with m inpts, sbject to the strctred normalized pertrbations (13); hence, E =, F = A, andg = B in (19) Since E and G in (19) need to be sqare and inertible, so to find the strctred real stabilizability radis, we consider the following system instead: ẋ = ( A + Ē ( Δ A F) x + B + Ē Δ BḠ) (2) where B = B ɛi is sqare, Ē =, F = A, and Ḡ = B R I,whereR R ɛi 2 (2 m) is an orthonormal matrix chosen so that each of the colmns of R is orthogonal to each of the colmns of B, andwhere ɛ > is chosen as small as possible (eg ɛ = 1 6 ) sch that Ē and Ḡ are inertible and not ill-conditioned In this case, the normalized real stabilizability radis of the pendlm system, denoted by r c,p (normalized), is eqal to the strctred real stabilizability radis of ( A, B, Ē, F, Ḡ) 3) Reslts: Table VI gies the absolte and normalized real stabilizability radis of a -link inerted pendlm for = 1, 2,,7 Here, we see that as the nmber of links in the pendlm system increases, the system loses stabilizability robstness TABLE VI REAL STABILIZABILITY RADIUS OF PENDULUM r c,p(absolte) r c,p(normalized) Sig Fig Reqired VI MULTI-INPUT MULTI-LINK PENDULUM SYSTEM We now repeat a similar stdy for the same mlti-link inerted pendlm system (4), bt with torqe inpts at all of the links of the system (see Fig 4) A Nonlinear and Linearized Eqations of Motion Fig 4 illstrates a -link pendlm system with control inpts, where a separate torqe is applied to each of the links piots Defining the inpt ector, = T 1, and linearizing abot the ertical eqilibrim point, (x, ) = (, ), we obtain the following θ 1 θ 2 m 1 1 m 2 Fig 4 Model of a mlti-inpt mlti-link inerted pendlm with links and inpts linearized model: ẋ = (M L ) 1 x + M a (M L ) 1 M b y = 1 x (21) where M b =diag ( l l ) B Reslts The reslts for the nonlinear simlations are shown in Table VII IX The reslts for the real stability and stabilizability radis are shown in Table X XII TABLE VII MAX INITIAL ANGLES (IN DEGREES) FOR MULTI-INPUT PENDULUM Initial Configrations Aligned Bottom-only Alternating TABLE VIII MAX DISTURBANCE FOR THE MULTI-INPUT PENDULUM θ l 2 2 l m b max VII CONCLUSIONS A mlti-link inerted pendlm system with links and controlled by a single torqe inpt is stdied in this paper Sch systems hae the property that when 4, it is not possible to experimentally stabilize them, in 1818

6 TABLE IX ALLOWABLE UNCERTAINTIES FOR MULTI-INPUT PENDULUM Parameter Accracy r LM (Sig Fig) TABLE X REAL STABILITY RADIUS OF MULTI-INPUT PENDULUM r s,p(absolte) r s,p(normalized) Sig Fig Reqired spite of the fact that the linearized model of these systems is controllable for all ales of It is conjectred that the main reason for this difficlty is that as the nmber of links increases, the pendlm system s real stabilizability radis becomes excessiely small, which implies that no practical controller exists which can experimentally stabilize sch a system In contrast, it is shown that if the -link inerted pendlm system has torqe inpts, then no sch difficlty arises TABLE XI REAL STABILITY RADIUS OF MULTI-OUTPUT CONTROLLER r s,k (absolte) r s,k (normalized) Sig Fig Reqired TABLE XII REAL STABILIZABILITY RADIUS OF MULTI-INPUT PENDULUM r c,p(absolte) r c,p(normalized) Sig Fig Reqired REFERENCES 1 K Frta, T Ochiai, and N Ono, Attitde control of a triple inerted pendlm, International Jornal of Control, ol 39, pp , P J Larcombe, On the control of a two-dimensional mlti-link inerted pendlm: the form of the dynamic eqations from choice of coordinate system, International Jornal of Systems Science, ol 23, no 12, pp , K G Eltohamy and C Y Ko, Nonlinear generalized eqations of motion for mlti-link inerted pendlm system, International Jornal of Systems Science, ol 3, no 5, pp , May P J Larcombe, Towards the theoretical controllability of a friction damped mlti-link inerted pendlm: the qadrple link system, N- link inferences, in International Conference on Control, o, March 1994, pp V A Tsachoridis and G A Medrano-Cerda, Discrete-time H control of a triple inerted pendlm with single control inpt, IEE Proceedings - Control Theory and Applications, o46, no 6, pp , No K G Eltohamy and C Y Ko, Real time stabilisation of a triple link inerted pendlm sing single control inpt, IEE Proceedings - Control Theory and Applications, o44, no 5, pp , Sept S Lam and E J Daison, Real stability and stabilizability radii of the mlti-link inerted pendlm system, Uniersity of Toronto, Tech Rep 52, May 25 8 G H and E J Daison, Real controllability/stabilizability radis of LTI systems, IEEE Transactions on Atomatic Control, ol 49, no 2, pp , Feb 24 9 Q Li, B Bernhardsson, A Rantzer, E J Daison, P M Yong, and J C Doyle, A formla for comptation of the real stability radis, Atomatica, ol 31, no 6, pp , 1995 APPENDIX The following extension to 8 is made in 7 Definition A1 (Strctred Real Stabilizability Radis): Gien the stabilizable LTI system ẋ = Ax + B (22) where A R n n, B R n m, and gien nonsinglar matrices E R n n, F R n n, and G R m m, the strctred real stabilizability radis of (A, B, E, F, G) is defined as: r c (A, B, E, F, G) =inf { σ(δ A, Δ B ) : Δ A R n n, Δ B R n m and (A + EΔ A F, B + EΔ B G) is nstabilizable} (23) The following reslt is obtained in 7: Theorem A1: r c (A, B, E, F, G) = min μ c (W) (24) s C + where ( ) μ c (M) = inf σ RM γim 2n 1 γ (,1 γ 1 IM RM (25) σ 2n 1 ( ) denotes the second largest singlar ale of ( ), and W = E 1 A si, B 1 F (26) G 1819