A Novel Model Predictive Control Strategy for Ane Nonlinear. Control Systems. John T. Wen. Center for Advanced Technology

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1 999 RPIC (Mar del Plata (Arg)) A Novel Model Predictive Control Strategy for Ane Nonlinear Control Systems Fernando Lizarralde Dept. of Electronic Eng./EE Federal University of Rio de Janeiro Rio de Janeiro, RJ 2945/97, Brazil fernando@coep.ufrj.br John T. Wen Center for Advanced Technology Rensselaer Polytechnic Institute Troy, NY 28 wen@cat.rpi.edu. Liu Hsu Dept. of Electrical Eng./COPPE Federal University of Rio de Janeiro Rio de Janeiro, RJ 2945/97, Brazil liu@coep.ufrj.br Abstract This paper considers the stabilization problem for control-ane nonlinear systems. A gradient-based numerical method is proposed to solve this problem. A receding horizon strategy, from Model Predictive Control, is used to yield a feedback control law. We show that, under certain weak assumptions, the closed-loop system is asymptotically stable. The full swing-up of the cartpole system shows the possibility of considering fast nonlinear systems. Keywords: Nonlinear Systems. Model Predictive Control. Newton method. Redecing Horizon. Introduction The feedback stabilization of control-ane nonlinear systems has received considerable attention during the last years. Past approaches range from the classic openloop linearization about an equilibrium followed by linear feedback stabilization control, to nonlinear feedback linearization and even to fuzzy control. Our approach is based on a class of iterative algorithm which was developed for robot path planning [3, 3, 4]. We have modied the original open-loop type of strategy into a feedback implementation. The key idea was to combine planning and control so that they are performed simultaneously rather than start control action only after the planning is completed. The resulting feedback controller reveals itself similar to Model Predictive Control (MPC), with the important distinction that the number of computation is xed in each control interval. Thus, in contrast of the slow process control usually associated with MPC, the strategy proposed here seems more adequate to be used for the control of relatively fast systems. We have successfully applied the approach to driftless nonlinear systems including nonholonomic systems [7, 8]. In this paper, we extend our approach to ane nonlinear control systems. Under certain Lipschitz condition on the system and non-singularity assumption for certain gradient operator, we can guarantee global exponential stability of the predicted error and global asymptotic stability of the actual error. To demonstrate the eectiveness of the controller, we apply it to the full swing-up of the cartpole system. 2 Preliminary Results Consider a time-invariant nonlinear system: _x = f (x) + mx i= f i (x)u i = f (x) + f(x)u () where f i is a smooth vector eld, x 2 IR n, (u u m ) 2 U U, where U is the space of admissible control functions, which consists of piecewise constant functions which are piecewise continuous. We will assume that the state is fully measurable and system () is strongly accessible []. Here, we will consider the design of controllers that stabilize system about a desired state x d, such that the pair (x d ; u ) characterizes an equilibrium condition, i.e., f (x d ) + f(x d )u =. A possible way to solve the control problem is to adopt an optimal control point of view and to regard () as a nonlinear functional mapping of the input function u T 2 L 2 m[t; t + T ) to the nal state x(t + T )2IR n : x(t + T ) = T (x(t); u T ) (2) which gives the state at time t + T starting from time t at the state x and applying the input function u T := fu(t) : t 2 [t; t + T )g. The problem is to nd a control input u T such that x(t+ T ) = x d and it minimizes a cost function J(x; u T ; T ), where x is the state trajectory generated by u T. The optimization problem could also consider constraints on u(t) and/or x(t). For general nonlinear systems, numerical methods are used to determine u T that minimize J. Classical iterative techniques [2, 5] consider x()=x, start from an initial guess u T (), and attempt to improve recursively on this initial control, obtaining a sequence fu T (); u T (); : : :g, so that J(u T ()) > J(u T ()) > : : :. In the case of considering J(u T ) = jj T (x ; u T )? x d j 2 the problem can be solved using the Newton method [2]. This result can be derived in a continuous-time framework, considering the predicted error: e T () = e(t; x ; u()) = T (x ; u T ())? x d (3) and its derivative with respect to (iteration variable): de T d = r u T (x ; u T ()) du d : (4)

2 where the map r u T denotes the Frechet derivative of T (x ; ) with respect to u T [5]. If r u T (x ; u T ) is onto, then we can choose the following update law for u T () (see [4] for other choices): du T d =? r u T (x ; u T ()) y et () (5) where > and [] y denotes the Moore-Penrose pseudoinverse. The gradient operator r u T can be computed from the system () linearized about the path corresponding to u T [5]. A sucient condition for the convergence of this algorithm is r u T (x ; u T ()) be onto for all, or, equivalently, the time-varying linearized system about the path generated by each u T () is controllable [5]. Under this condition, substituting (5) into (4) yields de T d =?e T which implies the exponential convergence of e T () to zero, more exactly, jje T ()jj jje T ()jj exp?. The update law (5) can be implemented in iterative form using a discrete basis for the control function space U according to the following update law: u T ( + ) = u T ()? r u T (x ; u T ()) y et () where is an integer. In this case, we can prove that there exists > such that, if e T () 2 B() (a ball of radius ), the predicted error e T () converges quadratically to zero. A globally convergent algorithm can be obtained combining this update law with a line search over the parameter, e.g. using Armijo rule [5] (a bisection search). Others methods such as continuation/homotopy [] can be used to accomplish the same objective. The Armijo rule guarantees that the error e T () converges exponentially to zero if e T () 62 B() and quadratically whenever the error gets inside B(). The above approach renders only an open-loop controller. To ensure the system can actually follow the path, some type of feedback is needed. A common approach is to linearize the system about the desired open-loop path and design a time-varying linear controller to keep the system on the path. Due to the linearization, the resulting stability is in general only local in nature. 3 A Model Predictive Strategy In this section, we present a modication of the openloop iterative approach to render it in a feedback form. The main idea is to simultaneously perform the iteration on the variable and the execution of the control u(t). At each time, the current state, x(t) is used as the initial condition for the map T (; u T ) (3) and the gradient r u T (; u T ) (5). After the control function u T is rened with one Newton-step, the control at time t, based on u T, is executed to drive the system to a new state and the procedure repeats. The Newton-step guarantees that the predicted error is strictly decreasing, therefore we can show the convergence of the state to the desired value. To describe the above procedure analytically, it is convenient to consider the system discretized in time. Denote the control vector at the kth time interval as u M (k) = [u M (k=k); ; u M (k + M? =k)] (6) where u M 2IR mm, M is an integer which denes the time window T =Mh (h: sampling period) and u M (i=k)2ir m (i k) is used to indicate that it is a prediction of u(i) based on measurements available at time k. The M-step ahead predicted state error is dened by: e M (k) = ^x(k + M=k)? x d = M (x(k); u M (k))? x d (7) where x(k + M=k) = M (x(k); u M (k)) is the predicted state at time k+m based on measurements available at time k. The Newton-step update of u M (k) is given by: v M (k) = u M (k)? r u M (x(k); u M (k)) y em (k) (8) A receding horizon strategy is dened by considering m z } { u(k) = v M (k=k) = e T n v M (k) (e T n = [ ]), the rst element of v M (k), which is applied to the actual system () yielding: x(k + ) = (x(k); v M (k=k)) (9) The receding horizon strategy is sketched in Figure. Then, the control vector is updated and shifted forward by one step: u M (k + ) = Gv M (k) + F u () where G 2 IR mmmm and F 2 IR mmm are dened as G= m(m?)m I m(m?) ; F mm = m(m?)m mm(m?) I m Note that, in (), the last element of u M (k + ) is lled with u. This choice will guarantee that x(k)! x d (c.f. Corollary ). In the case of f (x d ) = or f (driftless systems), u can be chosen to be zero. 3. Linear Systems Before getting into the stability analysis of the proposed model predictive method, it is worth to consider a discrete linear system, x(k + ) = x(k) +?u(k) () where x2ir n and u2ir m. For the sake of simplicity, we will consider the SISO case (m=) and the stabilization about the origin, i.e., x d = and u =. The end-point mapping of () is given by: ^x(k+m=k)= M (x(k); u M (k)) = M x(k) + C o u M (k) (2) 2

3 x d PSfrag replacements ^x(m + 3=3) = M (x(3); u M (3)) x(2) x(3) ^x(m + 2=2) = M (x(2); u M (2)) x() x() Figure : Model Predictive Strategy ^x(m + =) = M (x(); u M ()) ^x(m=) = M (x(); u M ()) where C o = [ M??; M?2?;?;?] is the controllability matrix. Furthermore, one has that the gradient of M is precisely given by: r u M (x(k); u M (k)) = C o The full rank of r u M means that the discrete time system () is controllable. The controllability of a sampled system is preserved if for a given sampling interval h, h( l? j ) (l = ; ; n; j = ; ; n) is not of a form 2ki for any pair of distinct eigenvalues l;j of matrix A, where k is an integer and i is the complex variable [4]. From (8), the update law is dened by: v M (k) = u M (k)? C y o ^x(k + M=k) =?C y o M x(k) + (I?C y oc o )u M (k)(3) and the receding horizon strategy is dened by u M (k + ) = Gv M (k) (4) where G is dened as in (). Finally, the control applied to the plant is given by u(k) = v M (k=k) = e T nv M (k) (5) Substituting (2) and (3) in (5), we have that the feedback law is dened by u(k) =?e T n Cy o M x(k) + e T n (I? Cy o C o)u M (k) (6) For = and M = n, C o is a square matrix and consequently (I? C o? C o ) =, therefore the feedback control law is given by u(k) =?Kx(k), where K =?e T n C o? M x(k) is the Ackerman formula, which places all the closed-loop poles at the origin, dening a dead-beat controller. For = and M > n, the update law (3) denes a design that cancel the predicted state at k =. After some algebraic manipulation, one has that the predicted state ^x(m + =) is given by: ^x(m + =) = M x() + C o u M () (7) = M+ x() + C o v M () (8) therefore, the update law (3) with = imposes that ^x(m + =) =. Considering u M (), v M () with = is optimal in the sense that minimize the control energy. In fact, the second element of (3) expands the null space of C o, thus dierent choices of u M () will not change the nal error in (7), but the transient response. Furthermore, ^x(k + M=k) = for k, which implies that v M (k) = u M (k) (k ) and from (4) we have that u M (k + ) = Gu M (k). Since G has M poles at the origin, u M (k) will be zero for k M. This pseudo dead-beat control is far less exigent, however, it could still present large control signals. One way to smooth the control is to consider 2(; ], which relax the condition ^x(m+=)= to the following contraction: ^x(k + M + =k + ) = (? )^x(k + M=k) In [6] is proposed a modication to the update law (3): v M (k) = u M (k)?c y o[^x(k+m=k)?(? )^x(k+m?=k)] (9) with 2 [; ], which results in a contraction independent of : ^x(k + M + =k + ) = (? )^x(k + M=k) For M = n, u(k) =?Kx(k) with K = e T nc o? n? (? I), which place n? poles at the origin and the remaining pole at. In [9] is presented a related approach for linear systems, where a state feedback linear controller is designed to satisfy a contraction ^x(k + M=k) = x(k), and u M () is chosen in order to satisfy a linear quadratic criterion. 3

4 4 Stability Analysis In this section, we will present the stability analysis of the proposed model predictive algorithm. Instead of the the update law (8), we will consider a law similar to (9): v M (k) = u M (k)? r u M (x(k); u M (k)) y (k) (2) where (k) := e M (k)? (? )e M? (k), with e M (k) and e M? (k) are dened as in (7). Consider rst the following assumptions for the system () and its end-point map (7): Assumption There exist i >, (i = ; : : :; n) such that jf i (x)? f i (y)jj i jx? yjj for all x; y 2 IR n. Assumption 2 There exist > and > such that jjf (x) + f(x)u jj jjxjj +, for all x 2 IR n. Assumption 3 The gradient r u M (x(k); u M (k)) is full rank for all k, and there exists m f > such that [ru M ] y < mf. Assumption 3 is equivalent to the control input history u M (k) being nonsingular at every time k. In practice, the possibility of encountering singular controls is generically rare and in the unlikely case that one is encountered, a generic loop can be appended to render r u M nonsingular [4]. Another technique is to augment the update law using a basis which spans the null space of r u M [2]. Considering e M? (k + )= M? (x(k + ); u M? (k + ))?x d from (9), () and the property: j ( i (x; u i ); u j ) = j+i (x; u j+i ), we have that e M? (k+) = M (x(k); v M (k))?x d (2) From the Taylor series expansion of M (x(k); v M (k)) about x(k) and u M (k), we have that e M? (k+)=e M (k)+r u M (x(k); u M (k))v M (k)+o(v M ) where v M (k) = v M (k)?u M (k), and o() takes into account the high order terms of the Taylor series. Thus, from the update law (2) and assumption 3, e M? (k + )=(? )e M (k) + (? )e M? (k) + d((k)) where d((k)) is a term of order (k). Considering =, the predicted error is governed by the following dierence equation: e M? (k + ) = (? )e M? (k) + d((k)) (22) We will show that the predicted error e M (k) converges exponentially to zero: Theorem Consider the error equation (22) and that assumptions 2 and 3 hold. There exist >, 2 (; ] and M such that if e M () 2 B(= ), then e M? (k) and e M (k) converge exponentially to zero. Proof: Without loss of generality, we consider that the Euler approximation is been used for solving (k), i.e. (k) = h[f (e M? (k) + x d ) + f(e M? (k) + x d )u ] + e M? (k) (23) Thus, from assumption 2, it follows that j(k)j (h + ) je M? (k)j + h (24) Considering now the Lyapunov function V (k) = e T M? (k)e M?(k), the increment V = V (k+)?v (k) is given by V =?(? 2 )e T M?(k)e M? (k) + 2e T M?(k)d((k)) + d T ((k))d((k)) where =? 2 and d() satises lim jjd()jj jj! jj =. Therefore, for any >, there exists r > such that jd((k))jj < jj(k)j, for all jj(k)j < r; or equivalently, using (24), jjd()j < [h + ] je M? (k)j + h ; (25) for jje M? (k)jj < r?h h+. Hence, V?[? 2?2(h? )? 2 (h? ) 2 ] je M? (k)jj 2 +2h jje M? (k)jj + (h ) 2 (26) Thus, for h suciently small, a predicted horizon T, M = T=h, and 2 (; ), there exists some 2 (; ] such that, V? jje M? (k)jj 2 ; < < or equivalently je M? (k + )jj (? ) jje M? (k)jj Then, from the Lyapunov theorem, the predicted error e M? (k) converges exponentially to zero. Now, re-writing e M (k) as: e M (k) = ( M? (x(k); u M? (k)); u )? x d and since M?! x d as k!, we have that lim e M(k) = (x d ; u )? x d = k! where (x d ; u ) = x d. Thus, we conclude that e M (k)! as k!. The result obtained in Theorem is semi-global, i.e., we can arbitrarily increase the stability domain, B(= ). This would imply in slow rate of convergence, since should be chosen close to zero. A way to obtain a global result is to perform a line search over, for example implementing the Armijo rule [5], which guarantee that e M (k) suciently decrease reducing the parameter. Moreover, it can be show that the line-search will terminate in a nite number of iterations [5]. In order to prove that the actual state x(k) converges to x d, we will rst present the following auxiliary lemmas: 4

5 Lemma The control signal u(k) is uniformly bounded for all k. Proof: We have that u(k) = e T n v M (k), thus, u(k) is bounded by jv M (k)jj. From (2) and (), v M (k) = Gv M (k?)+f u?[r u M ] y (k). Then, from Theorem, (k) converges exponentially to zero, therefore, assumption 3 and since G denes a BIBO stable system, we have that v M (k) uniformly satises: jju(k)jj k jv M (k)jj k 2 + k 3 m f jj()jj ; As a consequence of Lemma, equation () has the RHS uniformly Lipschitz (in x), if f and f are (assumption ), and it is piecewise continuous in t. This result guarantees the existence and uniqueness of solution. Lemma 2 All the element of the u M (k) tend to u as k!. Proof: Consider the update law (2). From Theorem, (k) tends to zero, then if assumption 3 holds, we have that v M (k) = v M (k)? u M (k)!, i.e, 8k The exponential rate of convergence of the predicted error implies that there should be some stability robustness with respect to modeling error. This type of convergence may allow for parameter adaptation if the model dependent portion can be written in a linear-in-the-parameter form. 5 Simulation Results Here, we will consider that a cart of mass m c = has a uniform pendulum of mass m p = : and length l = :5 pivoted on its top and is controlled by applying an input force u(t) (Fig. 2). The system is described by: (m c + m p )y? m p lcos() + m p lsin() _ 2 = u J? m p lcos()y? m p lgsin() = where J = 4=3m p l 2 is the pendulum inertia, g is the gravity, is the angle of the pendulum with respect to the vertical position and y is the cart displacement. This system can be written in ane form () with state v M (k) = u M (k) + "(k) where "(k) is a generic exponentially decreasing term. Thus, equation () becomes u M (k) = Gu M (k) + F u + "(k) Since matrix G denes a dead-beat system (i.e, all poles of G are placed at the origin), we obtain (i = ; ; M? ) u M (k + i=k) = u as k! PSfrag replacements u Figure 2: Inverted Pendulum y The following corollary shows that the actual error (x(k)? x d ) also converges to zero. Corollary Consider that assumptions 2-3 hold. Then, the actual system state, x(k), converges to x d as k!. Proof: Consider M (x d ; u M (k)). Then, assuming that, for a given initial condition, u M is a piecewise continuous function of time, from the ODE uniqueness of solutions [, Teorema 4.4], we have that: jjx(k)?x d jj j M (x(k); u M (k))? M (x d ; u M (k))jj e LMh (27) where L is the Lipschitz constant f +fu. From, Lemma 2, we obtain lim k! M(x d ; u M (k)) = M (x d ; u ) = x d and from Theorem, M! x d as k!, then, we have that lim jjx(k)? x djj lim jje M(x(k); u M (k)jj e LMh k! k! which implies lim k! jx(k)? x d j =. x T =[; ; _ y; _y]. The predicted trajectory generated by the control signal u M was calculated using the Euler approximation, while we apply the control to a real plant which is simulated using a 4th order Runge-Kutta algorithm. The parameter (2) is found to guarantee a sucient decrease in je M? (k)jj, jje M? (k)jj (? ) jje M? (k)jj ( = e?4 ), reducing the size of as: new = :5 old. The state and control trajectories are shown in Figure 3 for h = :2, M = 5, u M (), and initial and desired congurations given by: x T () = [ ] and x T d = [ ] respectively. The cart goes through an intuitive move: left and then right to swing up the pole. The control force is very large, however. When the discretization interval is increased (resulting in larger approximation error), convergence still results. When this interval is increased further, there is no longer convergence. In order to avoid very large control force, we can use a penalty function method to impose a hard bound on the control. The penalty function is dened as: g(u)=(?e r:c(u) ) 2 5

6 Unconstrained Case: ( ) pole angle (rad); ( ) cart position (m) 4 Constrained Case: ( ) pole angle (rad); ( ) cart position (m) Control signal u Control signal u Figure 3: State and Control Trajectories. Unconstrained case. h = :2; M = 5; u M () Figure 4: State and Control Trajectories. Constrained case. h = :2; M = 5; u M (). where r >, > and juj? Umax if juj U c(u) = max otherwise Evaluating g(u) along u M (k): z(k) = X M? i= g(c(u M (k + i=k)); the input constraint can be embedded in the algorithm augmented the error vector in (2) to include z(k). We imposed the limit of 2N. The result is shown in Figure 4 for = and r =. When the limit is tightened to N, the algorithm no longer converges. It is well-known that there exist several solutions to the full swing-up and stabilization problem. However, the application above illustrates the ecacy and simplicity oered by the proposed approach in solving this benchmark example. In contrast to the solutions found elsewhere, here no switching of the control law, based on some qualitative analysis of the solutions, is required. Indeed, straightforward application of the method is made with no more than the system's dierential equation and the constraints specication are required to realize the control. 6 Conclusions We presented an extension of the nonlinear model predictive type of control strategy that we have previously proposed for driftless nonlinear systems. We have shown that under certain Lipschitz and non-singularity assumptions, the predicted error converges exponentially to zero and actual state error converges asymptotically to zero. Simulation results involving the full swing-up of a cartpole system show encouraging performance. References [] E. Allgower and K. Georg, Numerical Continuation Methods. Springer-Verlag, 99. [2] A. Bryson and Y. Ho, Applied Optimal Control. Braisdell Pub. Company, 969. [3] A. Divelbiss and J. Wen, \A path space approach to nonholonomic motion planning in the presence of obstacles," IEEE Trans. on Robotics and Automation, vol. 3, pp. 443{45, June 997. [4] R. Kalman, Y. Ho, and K. Narendra, \Controllability of linear dynamical systems," Contributions to Dierent Equations, vol., pp. 89{23,

7 [5] C. Kelley, Iterative Methods for Linear and Nonlinear Equations. SIAM, 995. [6] F. Lizarralde, Stabilization of Ane Nonlinear Control Systems by a Newton type Method. PhD thesis, Programa de Eng. Eletrica, COPPE/UFRJ, Rio de Janeiro, Sept [7] F. Lizarralde and J. Wen, \Feedback stabilization of nonholonomic systems in presence of obstacles," in Proc. IEEE Int. Conf. on Robotics&Automation, (Minniapolis), pp. 2682{2687, 996. [8] F. Lizarralde, J. Wen, and D. Popa, \Feedback stabilization of nonholonomic systems," in 996 Proc. Conf. on Information Sciences and Systems (CISS'96), (Princeton, NJ), 996. [9] A. Malmgren and K. Nordstrom, \Optimal state feedback control with a prescribed contraction property," Automatica, vol. 3, no., pp. 75{756, 994. [] R. Michel and A. Miller, Ordinary Dierential Equation. Academic Press, 982. [] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems. New York, NY: Springer- Verlag, 99. [2] D. Popa and J. Wen, \Characterization of singular controls for nonholonomic path planning," in Proc. 3 th IFAC World Congress, (San Francisco), pp. 6{66, 996. [3] S. Seereeram and J. Wen, \A global approach to path planning for redundant manipulators," IEEE Trans. on Robotics and Automation, vol., no., pp. 52{6, 995. [4] E. Sontag, \Control of systems without drift via generic loops," IEEE Trans. on Automatic Control, vol. 4, pp. 2{29, July 995. [5] E. Sontag, Mathematical Control Theory. Springer- Verlag, 2 ed.,

ENERGY BASED CONTROL OF A CLASS OF UNDERACTUATED. Mark W. Spong. Coordinated Science Laboratory, University of Illinois, 1308 West Main Street,

ENERGY BASED CONTROL OF A CLASS OF UNDERACTUATED MECHANICAL SYSTEMS Mark W. Spong Coordinated Science Laboratory, University of Illinois, 1308 West Main Street, Urbana, Illinois 61801, USA Abstract. In