Safe Manual Control of the Furuta Pendulum

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1 Safe Manal Control of the Frta Pendlm Johan Åkesson, Karl Johan Åström Department of Atomatic Control, Lnd Institte of Technology (LTH) Box 8, Lnd, Sweden PSfrag replacements Abstract This paper deals with manal control of nstable systems, sbject to control signal satration. In particlar we will consider the Frta pendlm, where the problem is to control the orientation of the arm manally while stabilizing the inverted pendlm. This paper presents an analysis, which leads to insight into the problem as well as a control strategy. This control strategy has been implemented on the real Frta pendlm. Aspects of the implementation as well as experimental reslts are discssed in the paper. r Stabilizing Control Reference Following Unstable Plant Fig.. An illstration of the problem formlation dealt with in this work. x I. INTRODUCTION Many systems are controlled by a combination of atomatic and manal control, aircrafts are typical examples where stability agmentation systems have been sed for a long time to simplify the task of the pilot. The combination of manal and atomatic control is particlarly crcial for nstable systems with actator constraints. A severe problem with sch systems is that the feedback loop is broken when the actator satrates and the system may reach a state where stability cannot be recovered. The system can be driven to sch a state nintentionally by manal control or by a combination of manal control and distrbances. The Swedish Fighter JAS 39 Gripen is an aircraft which is nstable in some flight conditions, in order to achieve high performance. The flight control system is mission critical becase in some flight conditions the nstable mode is so fast that a pilot cannot stabilize the system. The flight control system shold ths flfill the dal task of stabilizing the aircraft withot restricting the maneverability nnecessarily. The satration is a rate satration cased by the hydralic actators driving the control srfaces, see [8]. The essence of the problem can be captred in the following formlation. Consider an nstable system with actator satration, see figre. Find a control strategy that stabilizes the system and provides facilities for manal control. The strategy shold be sch that the pilot can maniplate the system easily bt withot driving the system nstable. In this paper we discss a problem of this type, namely stabilization of a Frta pendlm, which is a pendlm hinged on a rotating arm, see []. The tasks are to keep the pendlm stabilized in the pright position while permitting manal control of the arm. These tasks are conflicting becase both have to be done sing the same control variable, the torqe on the This work was spported by the Swedish Research Concil for Engineering Sciences, grant TFR pendlm. Since the torqe is limited, manal control cold drive the torqe to satration. When this happens it is no longer possible to stabilize the inverted pendlm. This problem is similar to the aircraft problem. In both cases we have to stabilize an nstable system. The constraints are, however, different becase there is a rate satration in the aircraft. The problems associated with control system constraints are well docmented in the literatre. Ideas from [9], [7] and [] will be sed in the analysis and controller design. This article gives contribtions on the translation of a theoretically motivated controller into a working implementation. II. PRELIMINARIES In this section we will define the nonlinearity of interest, the satration. Frther a mathematical model for the Frta pendlm will be presented. The model is based on the derivation in [3], only the reslts are given here. Finally a linear controller design will be performed. The linear controller will be needed in the nonlinear control design. A. The satration nonlinearity The satration fnction that will be sed in the following is assmed to be defined by µ x < µ σ µ (x) = x µ x µ µ x > µ B. Pendlm model Consider the Frta pendlm in fig. Let the length of the pendlm be l, the mass of the weight M, the mass of the pendlm m, its moment of inertia J and the moment of inertia for the arm J p. The length of the arm is r. The angle of the pendlm,, is defined to be zero when in pright position and positive when the

2 for ϕ =. Notice that the linearized model is parameterized with respect ϕ. The reason for this is that or objective is to perform velocity control of the Frta pendlm arm. This means that we will force this state to have different vales dring the control seqence, and in particlar, ϕ cannot assmed to be zero. Since this term is also very inflential on the reslting linearized model, it will be taken into accont in the design procedre. x y z ϕ Fig.. The Frta pendlm pendlm is moving clockwise. The angle of the arm, ϕ is positive when the arm is moving in conter clockwise direction. Frther, the central vertical axis is connected to a DC motor which adds a torqe proportional to the control signal. By introdcing the coefficients a = J p + M l b = J + M r + mr c = M rl d = lg(m + m/) the eqations of motion for the Frta pendlm may be written a aϕ sin cos + c ϕ cos d sin = c cos c sin + a ϕ sin cos +(b + a sin ) ϕ = σ µ (). () The coefficients for the pendlm sed in the experiments are: l =.43 m r =.35 m M =. kg J =.5 kgm J p =.9 kgm m =. kg As mentioned above, a linear model of the system will be needed for controller design in the following. Introdction of the state vector x = [ x x x 3 ] T = [ and linearization of the system () arond gives where ẋ = x = [ ϕ. α γ x + ] T β δ ϕ ] T σ µ () () α = abϕ + bd = 3.3 β = c ab c ab c = 7. γ = ac ϕ cd ab c =.588 δ = a ab c = 9 C. A Gain Schedled Controller In the following sections a nonlinear controller will be designed. In this procedre a linear controller that stabilizes the pendlm system withot the satration nonlinearity is assmed. Since all states variables are available as measrements from the real pendlm, state feedback will be sed for this prpose. However, a critical featre of the linearized model is that it is highly dependent on the arm velocity ϕ. As we can see, ϕ enters both α and γ as a qadratic term. Linear controllers designed for ϕ = performed very poorly and a gain schedled controller was therefore designed. This controller has the form = L( ϕ)(r x) (3) where r is the reference signal to be tracked and L( ϕ ) =(l l l 3 ). r is actally a vector; r T = ( x3 r), where the two zeros indicate the desired vales of the states and. The feedback gains L were calclated sing LQ design assming constant vales of ϕ ranging from to rad/s with a resoltion of rad/s. The vales were stored in a table. At rntime, linear interpolation was sed to obtain an approximation of the correct feedback vector. This approximation proved to work well, and moreover, it was essential to perform sccessfl control over the fll range of arm velocities. D. Friction There were also other practical complications. There was sbstantial friction in the motor driving the arm and the slip rings. The friction gave a significant limit cycle oscillation. This is illstrated in the experiment shown in figre 3 where the limit cycle in the arm angle has an amplitde of abot.45 rad. Several methods for friction compensation are discssed in [6]. A friction compensator based on the Colomb friction model F f ( ϕ, v) = F + c ϕ > F + c ϕ =, > F + c ϕ =, F c < < F + c F c ϕ =, < F c F c ϕ < (4) was designed. An estimate of the friction force was calclated sing this model and added to the control signal compted from the control law 3. The control signal is ths v = L( ϕ)(r x) + ˆF f

3 [rad] [rad/s] answer this qestion, let s examine the phase plane of the system. When the control signal satrates, the system is described by ẋ = x ẋ = α x ± β µ. ag replacements ϕ [rad] [g] t [s] ϕ [rad/s] t [s] Fig. 3. The effect of friction compensation. Notice the significant redction of the amplitde of the limit cycle for the arm angle, φ. Initially, there is now friction compensation. Friction compensation is activated at t = 3 s. where ˆF f is an estimate of the friction based on the model (4). In the following, it is assmed that friction compensation is sed bt we do not show it explicitly in the controller expressions to simplify the presentation. Figre 3 shows the drastic improvements obtained with friction compensation. Friction changes with the operating condition and friction compensation shold therefore be adaptive, see [6]. This was not done in the experiment, if necessary the parameters of the friction model were re-tned at the start of the experiments. III. PHASE PLANE ANALYSIS We shall now analyze the behavior of the linearized satration constrained system. This means that the control signal satration is the only nonlinearity in the model, which simplifies the analysis. A phase plane analysis will be performed, which will lead s to a design of a nonlinear controller. The prime objective of the controller is to stabilize the states x = and x =. Motivated by the fact that the state x 3 does not affect x and x we will start analyzing the sbsystem This system has two eqilibria: ( ) β µ (x eq, xeq ) = (p, ) = α, (6) ( (x eq, xeq ) = ( p, ) = β µ ) α,. (7) Frther we can conclde that the eqilibrim points are nstable, since the system matrix has the eigenvales λ = α and λ = α. To reveal the qalitative behavior of the system, consider the following expressions; = α ± β µ = α ± β µ d dt = α d dt ± β µ d dt = α ± β µ + C = α ( ± p) pα + C where the last expression describes a hyperbolic fnction. The hyperbolic axes are given by = ± α ( ± p) and corresponds to the stable and nstable eigenvectors of the system matrix given by the system (5). Figre 4 shows the behavior of the system (5). In the figre, the axes of the hyperbolic trajectories can be seen, as well as the lines, = µ (dashed) and = µ (dashed). We can also see that the state trajectories are stable for some initial conditions, bt diverge for others. That is, the phase plane is divided into one stable, and two nstable regions. In the plot, five different regions are marked, corresponding to different modes of operation. Ω : In this region the system operates linearly, i.e. the control law is = l x l x. and the trajectories converges to (x, x ) T = (, ). The region is determined by the lines = µ and = µ. ẋ = x ẋ = α x + βσ µ () Now, assme a control law on the form = l x l x. (5) Ω + : The control signal satrates, = µ. The region is bonded from below by the line = µ and from above by the line corresponding to the stable eigenvector of the right eqilibrim. The most significant characterization of this region is that the soltions are stable; trajectories starting in Ω +, converge to origin. The controller gains l and l are chosen to stabilize the system. When µ the system operates linearly, and is then stable. The interesting qestion is how the system behaves when the control satrates, i.e. > µ. To Ω : Eqivalent to Ω+, bt bonded by the lines = µ and the line corresponding to the stable eigenvector of the left eqilibrim.

4 6 4 Ω+ Ω Ω + Now, the location of the linear region Ω is critical. The expression for the satration planes that determine Ω are x = l (±µ + l x + c ) p q q x 4 6 p Ω Ω x Fig. 4. Phase plane for the constrained system (5). The solid crves are actal state trajectories. The dashed lines marks the satration limits. The important conclsion to be drawn from this phase plot, is that the state space is divided into a stable region (nshaded) and two nstable regions (shaded). Ω + : Also in this region the control signal satrates. The difference from Ω + and Ω is that trajectories in this region are nstable. Ω : Eqivalent to Ω+ ; nstable. To conclde, we have shown that there exists a region of attraction for the constrained system. An important observation is that satrating control inpts does not necessarily case instability, and that the stable region is significantly larger than the one implied by the satration limits. Let s now discss the conseqences of introdcing also the state x 3 = ϕ in the model, ẋ = x ẋ = α x + βσ µ () ẋ 3 = γ x + δσ µ () The control law is now assmed to be = σ µ ( l x l x l 3 (x 3 x r 3)). p (8) That is, we strive to achieve (x x x r 3 x 3)=( ), and in particlar, we want this relation to be tre in stationarity for constant x r 3. As stated above, the states x and x are not directly affected by x 3. That is, for any vale of x 3, the behavior of x and x when the control signal satrates is determined by the hyperbolic expression (8). Ths, the orthogonal projection of the hyperbolic axes on the x x plane will be as in figre 4. Frther the planes ( α s±p, s, t) constitte bondaries between important regions in the state space. Trajectories starting in the region between these planes may, bt does not necessarily, converge. Trajectories starting otside this sbset however, will never converge. This observation is essential, and will be sed in the following. where c = l 3 (x 3 x r 3 ) As we can see, the orthogonal projection of the linear region onto the x x plane will be translated in the x direction depending on the vale of c. The location of the linear region determines which of the two satrated trajectory sets that governs the behavior of the system. Moreover, if the projection of the linear region Ω overlaps with any of the regions Ω + or Ω the system may enter these nstable regions from where recovery is not possible. This is eqivalent to the regions Ω + and Ω disappearing. In order to prevent this, the existence of the regions Ω + and Ω shold always be enforced, so that the projection of the linear region never overlaps with the nstable regions. To conclde, if we scceed in restricting the movements of the projected linear region as describe above, the states of the system will not diverge for any combination of set point changes in x3 r. This reslt assmes that the projection of the initial conditions are contained in the region of attraction in figre 4, which is reasonable. IV. A CASCADED SATURATIONS CONTROLLER With this insight, we are ready to sggest a controller for the system (8), that prevents departre of the system for all possible changes in the reference vale x r 3. As above, we assme that the initial conditions are not nrecoverable. The troblesome term in the control law, and also the one that has to be restricted, is c. Motivated by this fact, we introdce the revised control law = σ µ ( l x l x + σ µ c ( c )). This control law offers the possibility to restrict c, in sch way that the regions Ω + and Ω always exist in the phase plane. This control strategy sing cascaded satrations is well known and docmented in the literatre. Work on similar problems has been made in []. An example of its applications is given in []. The controller strctre presented here, also has common characteristics with one sggested in [9]. It now remains to calclate admissible vales for µ c. The condition for garanteed stability is, as stated above, existence of the regions Ω + and Ω. The sitation may be illstrated as in figre 5. We will assme that the bondary line between Ω + and Ω+ and the line = µ are parallel. This can be achieved by introdcing a diagonal weighting matrix Q and a control signal weight R > in the LQ-design, and choosing the elements q = α q q = ρr α β ρ R, ρ α β

5 5 4 Ω + 3 Ω Ω q ε q Ω + Ω Ω + 3 Ω ε q Fig. 5. Calclation of admissible µ c. ε may be sed to obtain desired minimal size of the region Ω +. This choice of Q gives (l l )=(ρ α ρ) and satration lines parallel to the stable eigenvectors. This reslt is for the redced system (5), bt can be generalized to the fll system (8). Frther assme that we desire a safety margin for stability. In the figre 5 the margin is defined by ε q where q = β µ/ α and ε [, ]. This gives s an pper bond for µ c : µ c < l ( ε)q µ In figre 6 and figre 7 simlations of the linear constrained system controlled by the cascaded satrations controller. In the simlations, the satration limits were set to µ =.3 g. As we can see, the controller sccessflly stabilizes the the system, and provides reference following for the pendlm arm velocity. From the phase plot in figre 6 it is clear how the controller prevents the system from leaving the region of attraction. Also, the system does not enter the safety region defined by the dash-dotted lines. The step responses can be seen in figre 7 for different step sizes. The benefit from the cascaded satrations controller is obvios, if compared to the performance of a pre linear controller. In the simlations, stability is lost for some combinations of set point changes for the state x 3 when the linear controller is sed. When the cascaded satrations controller is engaged, the system will remain in the specified region in the state space. This reslt can readily be interpreted in terms of the control task, safe manal control. Given that the state of the system initially is contained in the stable regions, as described above, there exists no reference trajectory that will drive the system ot of these regions. That is, the controller garantees that stability is preserved, independently of the applied reference. The analysis presented in this section solves the problem for a linear plant, with constrained inpt. However, or analysis does not spport the same claims for the Fig. 6. A plane plot of two simlations. The solid crve shows the system behavior if the cascaded satrations controller is engaged. The dashed crve represents a simlation of the system controlled by a reglar linear controller. As we can see, the linear controller fails to stabilize the system. Ω Ω + Ω ϕ Step Responses Control Signal time [s] Fig. 7. The state trajectories for the simlations. The solid crves represents the case when the cascaded satrations controller is engaged, where as the dashed-dotted crves shows the response when a pre linear controller is sed. The control signal is normalized with respect to g. constrained nonlinear plant with gain schedled control. The region of attraction, which is the basis for the analysis will be different for the nonlinear pendlm model and it will be necessary to investigate the fll nonlinear problem. Having said this, we shall now explore the applicability of the cascaded satrations controller on the real Frta pendlm. Or ambition is to show that the controller designed for the linear model can, with some modifications, be made to work in practice on the real plant. V. EXPERIMENTS One of the objectives in this work has been to implement a well working control strategy for a real Frta pendlm. When applying a control strategy designed

6 ϕ Ω Ω + Ω Step Responses Control Signal time [s] Fig. 8. Reslts from experiments performed on the real Frta pendlm. The control signal is also here normalized with respect to g. for a linear model to a real plant as in or case, the potential difference between the model and the plant may be a sorce of problems. This application proved to be no exception. Varios problems, for example friction, noise and the inherent nonlinearities of the pendlm made the implementation far from straight forward. The friction problem was solved by introdcing friction compensation in the control loop. The compensation algorithm was based on Colomb friction with stiction and improved the performance considerably. Or main tool in dealing with the difference between the linear and nonlinear model has been gain schedling. This techniqe was sed for all controller parameters that depended on the system model. The procedre was described for the state feed back vector L above, bt also the satration limit µ c depends on the linearized model and shold ths be gain schedled. However, these modifications were not sfficient. The vale µ c also depends on the actator satration limit µ, which is different from the effective satration limits in the presence of friction and friction compensation. In order to perform experiments eqivalent to the simlations, the satration limits of the plant was set to µ =.5 g instead of µ =.3 g. The difference acconts for the friction. The design parameter ε was set to.5 in the experiments. With these modifications, the controller achieved the reslts in shown in figre 8. As we can see, the controller prodces slower step responses compared to the simlations, bt it keeps the pendlm in pright position while, in stationarity with constant x3 r, achieving x3 r x 3 =. Under the same circmstances, a pre linear state feed back controller failed to stabilize the system. The fact that the responses are slower may be explained by the fact that the modifications made to the controller reslted in a less aggressive controller. VI. RESULTS AND CONCLUSIONS In this paper, we have investigated the behavior of an nstable system sbject to manal control, in the presence of control signal satration, theoretically and practically. The inverted pendlm has served as an illstrative example for the analysis and design as well as the the experimental part of this work. The analysis of the constrained linearized pendlm model reslted in a controller design, that solved the stabilization problem. The nonlinear case, however, is not clear and wold be a natral extension of this work. The cascaded satrations controller was also implemented on a real Frta pendlm. The controller scceeded, after some modifications striving to make it less aggressive, in stabilizing also the real pendlm. VII. RELATIONS TO OTHER WORK In the corse of this work, we have also investigated other approaches to solve the problem of constrained control. In particlar, we have investigated the applicability of the methods presented in [5] and [4] which are based on very elegant analysis sing admissible sets and the reference governor. There were some difficlties in applying these ideas to the Frta pendlm, althogh they performed well in simlations. The main reason was de to the difficlty in compting the admissible sets accrately for the real pendlm which has friction and measrement noise. REFERENCES [] T. Brg, D. Dawson, C. Rahn, and W. Rhodes. Nonlinear control of an overhead crane via the satrating control approach. In Proc. of the 996 IEEE International Conference on Robotics and Atomation, pp , Minneapolis, Minnesota, 996. [] K. Frta, M. Yamakita, S. Kobayashi, and M. Nishimra. A new inverted pendlm apparats for edction. In IFAC Symposim on Advances in Control Edcation, Boston, MA, pp. 9 96, 994. [3] M. Gäfvert. Derivation of Frta pendlm dynamics. Report ISRN LUTFD/TFRT SE, Department of Atomatic Control, Lnd Institte of Technology, Lnd, Sweden, 998. [4] E. Gilbert, I. Kolmanovsky, and K. Tan. Nonlinear control of discrete-time linear systems with state and control constraints: A reference govenor with global convergence properties. In Proc. 33rd Conference on Decision and Control, pp , Lake Bena Vista, Florida, 994. [5] E. Gilbert and K. Tan. Linear systems with state and control constraints: The theory and application of maximal otpt admissible sets. IEEE Trans. on Atomatic Control, 36:9, pp. 8, 99. [6] H. Olsson, K. J. Åström, C. Candas de Wit, M. Gäfvert, and P. Lischinsky. Friction models and friction compensation. Eropean Jornal of Control, 4:3, pp , 998. [7] M. Patcher and R. Miller. Manal flight control with satrating actators. IEEE Control Systems, Febrary, pp. pp. 9, Febrary 998. [8] L. Rndqwist, K. Stål-Gnnarsson, and J. Enhagen. 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