Inverted pendulum swing-up with FSM controllers - Semantic Scholar
Fuzzy Sliding Mode Control for Inverted Pendulum Swing-up with Restricted Travel Mariagrazia Dotoli, Bruno Maione, David Naso and Biagio Turchiano Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari Via Re David, 200 – 70125 Bari – Italy e.mail: {dotoli,maione,naso,turchiano}@poliba.it Abstract Swinging up an inverted pendulum is a common benchmark task for the investigation of automatic control techniques. In this paper we introduce a new Fuzzy Sliding Mode (FSM) technique for swinging-up an inverted pendulum and controlling the connected cart, while minimizing the swing-up time, the cart travel and the required control action. The FSM technique adopted is based on a piecewise linear sliding manifold that is bent towards the far off zones of the pole phase plane, thus enabling the reduction of the control action. Further, in order to limit the cart overshoot two variable gains were inserted in the cart controller. We tested our technique both on a nonlinear model, including friction, and on a lab equipment: we report both on the upwards stabilization and swing-up. The pendulum upwards equilibrium point was made globally stable.
I. INTRODUCTION Swinging up an inverted pendulum is a common benchmark for the investigation of automatic control techniques. A pole, hinged to a cart moving on a limited track, is swung-up from its downwards stable equilibrium and balanced upwards by a horizontal force applied to the cart via a motor. The cart is simultaneously motioned to an objective position on the track. The control action is limited, since the motor presents a saturation effect. There have been many studies on this subject. For instance, Åström and Furuta [1] as well as Yoshida [7] solve the problem with energy-based techniques without controlling the cart motion; further, their methodologies are not tested on an application. Wei, Dayawansa and Levine [8] solve the problem with a nonlinear technique and apply their method to a lab rig, but they do not take into account the cart motion. The scope of this paper is to introduce a new Fuzzy Sliding Mode (FSM) technique for swingingup an inverted pendulum and controlling the connected cart, while minimizing the swing-up time, the cart travel and the required control action. The heuristic FSM technique adopted combines the Sliding Mode (SM) control methodology with fuzzy control algorithms [5, 3], using a piecewise linear sliding manifold that is bent towards the far off zones of the pole phase plane. We test our methodology both on a nonlinear model, including friction, and on a lab equipment. The paper is organized as follows. After a brief review of the SM methodology, in section 2 we introduce an FSM controller comprising a heuristic rule table based on phase plane considerations for a second-order dynamical system. In section 3 we outline a fourth-order nonlinear model for the inverted pendulum and design a modified FSM controller for the upwards stabilization of the pendulum and the cart control when the pendulum is initially close to its unstable equilibrium point. In section 4 we devise further changes of the FSM controller in order to swing-up the pendulum and report some results both for the simulated and real-world pendulum. Finally, some conclusions are outlined and suggestions for further research are driven. II. A FUZZY SLIDING MODE CONTROL TECHNIQUE Consider the following single-input second-order dynamic system expressed in a state-space companion form: x( t ) = f (x ( t )) + g(x ( t )) u ( t )
(1)
where x = [ x x ]T is the completely observable state vector, f(x(t)) and g(x(t)) are nonlinear functions and u(t) is the control input. Furthermore, consider a given desired trajectory xd(t) and the tracking error e(t)=x(t)-xd(t) of the state component x. The basic idea of SM theory is to force the system, after a reaching phase, to a sliding surface or switching line containing the operating point defined for (1) as: d s( x ( t )) = + λ e( t ) = e ( t ) + λe( t ) = 0 dt
(2)
where the sliding constant λ is strictly positive design parameter. The control action is designed in order to make the sliding surface attractive, i.e. such that the sliding condition holds [5]: s( x( t ))s(x ( t )) ≤ −η | s( x( t )) | .
(3)
It is well known that SM controls may exhibit high frequency switching (chattering) when the trajectory is near the sliding surface. The boundary layer SM technique [5] is efficient in solving this problem, but can compromise the tracking accuracy, since the tracking error magnitude directly depends on the boundary layer width. A further solution consists in combining the SM control methodology with fuzzy logic algorithms, thus relaxing the switching surface. The main advantages of the resulting FSM techniques are the small magnitude of the chattering effect and a low computational effort. The many different FSM control techniques can be roughly classified into two categories: fuzzy boundary layers and fuzzy Lyapunov functions. In the first and more popular approach the boundary layer is replaced by a fuzzy map, thus keeping the overall SM control structure and achieving stability and robustness guarantees similar to SM control. A further approach employs a fuzzy controller for an on-line tuning of the boundary layer width [2]. The second approach employs a fuzzy control law that completely substitutes the SM control laws, keeping the validity of the sliding condition (3). This method is generally more intuitive than the former and usually turns out to be less computationally intense. Finally, some FSM techniques can be placed midway between the two categories above. The most remarkable is probably the one by Palm [4]. A fuzzy control surface is designed, establishing a sort of variable gain in the state space. Note that in this approach the fuzzy controller input is either s(x(t)), or the normal distance sN(x(t)) from the sliding surface. If the input to the fuzzy controller is the latter, a further input is often the parallel component d(x(t)) of s(x(t)) along the switching surface. The technique by Palm is among the most interesting FSM algorithms, but requires considerable knowledge of the process and a remarkable computational effort. We propose a FSM control action governed by the non linear law: u = u fuzz (e, e , λ ) = u fuzz (s) .
(4)
The rule base stems from heuristic conditions in the phase plane, so that the overall control surface is akin to a SM control law with boundary layer but keeps the computational effort and chattering relatively low (the set of rules is remarkably simple). In order to derive the FSM controller rule base, note that when s(x(t)) is about zero the trajectory is close to the sliding manifold and the control action should be about zero. Furthermore, since the plant is of type (1), it holds:
if s is zero, then u is zero;
(ii)
if s is positive, then u
I. INTRODUCTION Swinging up an inverted pendulum is a common benchmark for the investigation of automatic control techniques. A pole, hinged to a cart moving on a limited track, is swung-up from its downwards stable equilibrium and balanced upwards by a horizontal force applied to the cart via a motor. The cart is simultaneously motioned to an objective position on the track. The control action is limited, since the motor presents a saturation effect. There have been many studies on this subject. For instance, Åström and Furuta [1] as well as Yoshida [7] solve the problem with energy-based techniques without controlling the cart motion; further, their methodologies are not tested on an application. Wei, Dayawansa and Levine [8] solve the problem with a nonlinear technique and apply their method to a lab rig, but they do not take into account the cart motion. The scope of this paper is to introduce a new Fuzzy Sliding Mode (FSM) technique for swingingup an inverted pendulum and controlling the connected cart, while minimizing the swing-up time, the cart travel and the required control action. The heuristic FSM technique adopted combines the Sliding Mode (SM) control methodology with fuzzy control algorithms [5, 3], using a piecewise linear sliding manifold that is bent towards the far off zones of the pole phase plane. We test our methodology both on a nonlinear model, including friction, and on a lab equipment. The paper is organized as follows. After a brief review of the SM methodology, in section 2 we introduce an FSM controller comprising a heuristic rule table based on phase plane considerations for a second-order dynamical system. In section 3 we outline a fourth-order nonlinear model for the inverted pendulum and design a modified FSM controller for the upwards stabilization of the pendulum and the cart control when the pendulum is initially close to its unstable equilibrium point. In section 4 we devise further changes of the FSM controller in order to swing-up the pendulum and report some results both for the simulated and real-world pendulum. Finally, some conclusions are outlined and suggestions for further research are driven. II. A FUZZY SLIDING MODE CONTROL TECHNIQUE Consider the following single-input second-order dynamic system expressed in a state-space companion form: x( t ) = f (x ( t )) + g(x ( t )) u ( t )
(1)
where x = [ x x ]T is the completely observable state vector, f(x(t)) and g(x(t)) are nonlinear functions and u(t) is the control input. Furthermore, consider a given desired trajectory xd(t) and the tracking error e(t)=x(t)-xd(t) of the state component x. The basic idea of SM theory is to force the system, after a reaching phase, to a sliding surface or switching line containing the operating point defined for (1) as: d s( x ( t )) = + λ e( t ) = e ( t ) + λe( t ) = 0 dt
(2)
where the sliding constant λ is strictly positive design parameter. The control action is designed in order to make the sliding surface attractive, i.e. such that the sliding condition holds [5]: s( x( t ))s(x ( t )) ≤ −η | s( x( t )) | .
(3)
It is well known that SM controls may exhibit high frequency switching (chattering) when the trajectory is near the sliding surface. The boundary layer SM technique [5] is efficient in solving this problem, but can compromise the tracking accuracy, since the tracking error magnitude directly depends on the boundary layer width. A further solution consists in combining the SM control methodology with fuzzy logic algorithms, thus relaxing the switching surface. The main advantages of the resulting FSM techniques are the small magnitude of the chattering effect and a low computational effort. The many different FSM control techniques can be roughly classified into two categories: fuzzy boundary layers and fuzzy Lyapunov functions. In the first and more popular approach the boundary layer is replaced by a fuzzy map, thus keeping the overall SM control structure and achieving stability and robustness guarantees similar to SM control. A further approach employs a fuzzy controller for an on-line tuning of the boundary layer width [2]. The second approach employs a fuzzy control law that completely substitutes the SM control laws, keeping the validity of the sliding condition (3). This method is generally more intuitive than the former and usually turns out to be less computationally intense. Finally, some FSM techniques can be placed midway between the two categories above. The most remarkable is probably the one by Palm [4]. A fuzzy control surface is designed, establishing a sort of variable gain in the state space. Note that in this approach the fuzzy controller input is either s(x(t)), or the normal distance sN(x(t)) from the sliding surface. If the input to the fuzzy controller is the latter, a further input is often the parallel component d(x(t)) of s(x(t)) along the switching surface. The technique by Palm is among the most interesting FSM algorithms, but requires considerable knowledge of the process and a remarkable computational effort. We propose a FSM control action governed by the non linear law: u = u fuzz (e, e , λ ) = u fuzz (s) .
(4)
The rule base stems from heuristic conditions in the phase plane, so that the overall control surface is akin to a SM control law with boundary layer but keeps the computational effort and chattering relatively low (the set of rules is remarkably simple). In order to derive the FSM controller rule base, note that when s(x(t)) is about zero the trajectory is close to the sliding manifold and the control action should be about zero. Furthermore, since the plant is of type (1), it holds:
if s is zero, then u is zero;
(ii)
if s is positive, then u
