Stabilization of constrained linear systems via smoothed truncated ellipsoids

Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Stabilization of constrained linear systems via smoothed truncated ellipsoids A. Balestrino, E. Crisostomi, S. Grammatico, A. Caiti Department of Energy and Systems Engineering, University of Pisa, Largo Lucio Lazzarino, 5622 Pisa, Italy e-mails: a.balestrino, a.caiti}@dsea.unipi.it. emanuele.crisostomi, grammatico.sergio}@gmail.com Abstract: Polyhedral Lyapunov functions are convenient to solve the constrained stabilization problem of linear systems as non-conservative estimates of the domain of attraction can be obtained. Alternatively, truncated ellipsoids can be used to find an under-estimate of the feasible region, with a considerably reduced number of parameters. This paper reformulates classic geometric intersection operators in terms of R-functions, leading to a new family of smooth Lyapunov functions. This approach can be used to smooth both polyhedral and truncated ellipsoids Lyapunov functions improving control performances, as shown in several benchmark examples. Keywords: Constrained linear systems, composed Lyapunov functions, R-functions.. INTRODUCTION A popular approach to solve many stability/stabilizability problems is based on the search of appropriate Lyapunov Functions (LFs)/Control LFs (CLFs). The main difficulty of this approach is that in many control applications it is not clear which class of functions is the most convenient to search for a suitable LF/CLF. An example is the stabilization problem of constrained linear systems, which is the topic of this paper, where the same problem can be solved with different levels of complexity and accuracy, depending on the choice of the class of candidate CLFs. The exact solution of the problem corresponds to finding the largest admissible controlled invariant region of the state space, where admissibility regards both state and control constraints (if any). An approximated solution is given by the largest controlled invariant ellipsoidal set included inside the feasible region, easily computed according to standard Linear Matrix Inequality (LMI) arguments [Boyd et al. (994)]. Such an approximated solution is obtained if quadratic LFs are exploited. On the otherhand,arbitrarilyaccuratesolutionscanbecomputed if Polyhedral LFs (PLFs) are used [Gutman and Cwikel (986)], [Blanchini (995)]. PLFs are more flexible than ellipsoidal functions, as they can be shaped to fit the exact feasible region; the polyhedral region is described by a linear set of inequalities like Fx and in [Blanchini and Miani (998)] it is also shown that the polyhedral function can be smoothed by restating the set of inequalities as Fx 2p, for an appropriate choice of p. The smoothed solution is more convenient because the induced LF becomes everywhere differentiable and gradient-based controllers can be used for obtaining timecontinuouscontrolsignals[petersenandbarmish(987a)]. Apparently, there is only one (major) drawback in the use of PLFs, i.e. the computational burden to find suitable LFs. More recently a trade-off solution has been proposed in the literature, where functions induced by truncated ellipsoidal sets are used as the basic class for the candidate LFs[O DellandMisawa(22)], [Thibodeau etal.(29)]. As shown in [O Dell and Misawa (22)], the advantage of using truncated ellipsoids is that on average a good approximation of the feasible region is computed with a considerablysmallernumberofparameters.in[thibodeau et al. (29)] it is further shown that the stabilizability problem using truncated ellipsoidal LFs can be casted in a quasi-lmifashion(i.e.asetofbilinearmatrixinequalities (BMIs) that can be reduced to a set of LMIs once some parameters are fixed). Note that the estimated admissible region is controlled invariant under a linear state feedback control law which is designed to satisfy the physical constraints and to maximize the size of the estimated feasible region. In this case gradient-based controllers can not be used because the LF is constructed by intersecting ellipsoidal with polyhedral sets and it is not everywhere differentiable. This paper provides two main contributions: first, the results of [Balestrino et al. (2)], where a new class of smoothed PLFs was introduced to stabilize Linear Differential Inclusion (LDI) systems, are generalized to include the case of constrained linear systems. The proposed smoothing technique is different from the conventional 2pnorm [Blanchini and Miani (998)] as the sublevel sets do not have the same shape everywhere inside the feasible region, but they become smoother near the origin. This property is expected to be convenient for control purposes as it is verified in several simulations. The second contribution is that the proposed smoothing technique can be extended to the case of CLFs induced by truncated ellipsoidalregions,andthereforegradient-basedcontrollers can be used also in the case of [O Dell and Misawa (22)] and[thibodeauetal.(29)].thisisanimportantnovelty Copyright by the International Federation of Automatic Control (IFAC) 6739

Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Table. Correspondence between logic functions and R-functions Boolean not and α or α R-composition r r +r 2 r 2 +r2 2 2αr r 2 2 2 2α r +r 2 + r 2 +r2 2 2αr r 2 2+ 2 2α as the 2p-norm can not be used to smooth the intersection of the ellipsoidal and polyhedral regions, and gradientbased controllers provide better control performances than the simple static state feedback ones, while preserving the same estimate of the admissible region. The proposed smoothing technique is easily derived from reinterpretingtheintersectionofellipsoidalandpolyhedral regions in the context of R-functions, which are realvalued functions that generalize the classic pointwise min and max operators. R-functions are described in the next section, while in Section 3 the proposed control law is introduced and motivated. Section 4 shows simulation results, where the derived control law is compared with the PCLFs of [Blanchini and Miani (998)] and the linear feedback control laws of [Thibodeau et al. (29)]. In the last section the main results are summarized and future lines of research are outlined. 2. R-FUNCTIONS FOR CONTROL APPLICATIONS R-functions were introduced in [Rvachev (982)] and more recently have been discussed in [Shapiro (27)]. Definition. (from [Shapiro (27)]). A function r : F n R n R is an R-function if there exists a Boolean function R : B n B, where B =,}, such that the following equality is satisfied: h(r(x,,...,x n )) = R(h(x ),h( ),...,h(x n )), where h : R B is the standard Heaviside function. The Boolean function R is also called the companion function of the R-function r. Informally, a real function r is an R-function if it can change one property (sign) onlywhensomeofitsargumentschangethesameproperty (sign). The parallelism between logic functions and R- functions becomes more evident when classic Boolean operators are recovered as described in Table. For instance, according to Table, the interpretation of the and composition is that the composed function is positive when evaluated in x if and only if both r (x) and r 2 (x) are positive. The result is obtained by exploiting the triangle inequality and the law of cosines, and it holds for all values of α [,] R [Balestrino et al. (29a)]. The terms at the denominator in Table are normalizing factors. Remark. It can be easily seen that when α =, r r 2 = minr,r 2 } and similarly r r2 = maxr,r 2 }. ThereisalsoanicegeometricinterpretationofR-functions as illustrated in the next example. Example. Consider the truncated ellipsoid obtained by intersecting the polyhedron Fx and the ellipsoid x Px, where.5.5.5 α =.5 2.5.5.5.5 2 x Fig.. The R-function r is calculated with α = for all of the R-intersections. [ ] [ ] 2/3 /3 F =, P =. 2/3 We consider functions r (x) = (2/3)x, r 2 (x) = + (2/3)x, r 3 (x) = (2/3), r 4 (x) = + (2/3), r q = x Px, where x = [x ] is the state vector. For convenience the functions have been normalized so that their maximum value is. Then we compute the intersection α 2 α 23 α 34 α r = r r 2 r 3 r 4 rq, according to the equation of Table, for arbitrary values of α,α ij [,]. The intersection function is positive inside the geometric intersection region, it is zero on the boundary, negative outside and its maximum value is at the origin. The sublevel sets of the function f = r (for α = α ij = ) are shown in Figure. The composed function is the smoothed intersection between the smoothed polyhedral function introduced in [Balestrino et al. (2)] and a quadratic function. Alternatively we can also compute the R-intersection between the (non-smoothed) polyhedral function Fx, r p = r r2 r3 f4, and the quadratic function r q = x Px, that is r p α rq. In this case, Figure 2 shows the sublevel sets of the corresponding fα = r p α rq with α = (truncated ellipsoid) and with α =. Remark 2. As previously mentioned, the intersection of a polyhedral region with ellipsoids has already been used as a candidate LF in [Thibodeau et al. (29)]. Such a LF is identical to the R-intersection of the polyhedral function andthequadraticonewithα =,asintheexampleshown on the left of Figure 2. Other control applications of the R-functions can be found in [Balestrino et al. (29a)] and [Balestrino et al. (29b)]. 3. CONSTRAINED STABILIZATION OF LINEAR SYSTEMS 3. Problem statement Let us consider the constrained stabilization problem of a linear system: ẋ(t) = Ax(t)+Bu(t) s.t. x(t) X R n, u(t) U R m (), 674

Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2.5 α =.5 α =.5.5.5.5.5 2.5.5.5.5 2 x.5 2.5.5.5.5 2 x Fig. 2. On the left the intersection between the polyhedral function and the quadratic one is performed with α =, while on the right with α =. where A R n n, B R n m. The control objective is to design a control law u(t) such that x(t) asymptotically converges to the origin, in accordance to the state and control input constraints. We assume that the set X is compact, convex and -symmetric, and that U = u R m : u }. (2) The first assumption is standard in constrained stabilization problems, while the second is only a simplificative assumption that does not affect the generality of the results. Polyhedral Lyapunov Functions Typically,thedrawbacksofusingQuadraticLFs(QLFs) are that the estimate of a positive controlled invariant set is restricted to ellipsoidal regions and that the control law is a linear state feedback. A PLF V : R n R is usually described by equation V (x) = Fx = max F i x }, (3) i where F R r n is a full column rank matrix, F i is the i th row of F, and sublevel sets have the shape of a polyhedron with 2r facets. PLFs outperform QLFs in terms of nonconservative estimates of a positive controlled invariant set, but the lack of differentiability causes some difficulties in the control synthesis. Smooth PLFs circumvent both difficulties, as they provide non-conservative estimates and enable the use of gradientbased controllers. In this framework, R-functions are used as an alternative way of expressing the PLFs, so that the parameter α can be used to tune the smoothness of the sublevel sets. The proposed smoothing technique is different from conventional ones, e.g. the 2p-norm of [Blanchini and Miani (998)] V 2p (x) = Fx 2p = 2p r (F i x) 2p, (4) i= because the sublevel sets become smoother close to the origin, which is usually convenient to improve control performances. The correctness of the proposed smoothing techniqueisprovedinsection3.2,whileacomparisonwith the 2p-norm is provided in Section 4 (Examples and 2). In the following, the smoothing procedure of [Balestrino et al. (2)] is summarized. It is assumed that a suitable polyhedral function for the stabilization problem is available,forinstanceitcanbefoundusingmethodsoutlinedin [Brayton and Tong (98)], [Blanchini (995)] or [Polanski (2)]. The corresponding R-function is computed as the intersection of the level set of the linear constraints forming (3). This is explained in the two-steps procedure (5)-(6) for clarity:,i (x) = ( F i x) α (+F i x), i =,2,...,r, (5) (x) = α r i=,i (x). (6) Thepolyhedronx R n : (x) = }isexactlythesame described by Fx =. The particular choice of the level set does not affect the generality of the approach, because the state can be appropriately rescaled. As previously remarked, (x) > x s.t. Fx < and max (x)} = () =. Therefore the associated x candidate (positive definite) LF is Vα(x) = (x). (7) Remark 3. Both function V 2p (4) for p + and Vα (7) for α = coincide with the polyhedral function V (3). Truncated ellipsoids The use of truncated ellipsoids provides an approximation of the maximal feasible set which is less conservative than the ellipsoidal approach, but computationally more convenient than polyhedral approximations, which might involve a very large number of vertices and facets. The use of truncated ellipsoids has been proved effective among others by [O Dell and Misawa (22)] and [Thibodeau et al. (29)], where linear feedback control laws are used. In [Thibodeau et al. (29)] the truncated ellipsoid function is described as V te (x) = max x Px, x Ci C i x }, (8) i where C i R n, for i =,...,r. The use of the max operator makes V te non differentiable. The same solution can be reformulated in terms of R- functions (i.e. via intersecting ellipsoids and polyhedral regions) and the parameter α can be used as a smoothing factor. Smoothing allows for the recovering of gradientbased controllers and average control performances are improved as it is shown in Section 4 (Examples 3 and 4). To perform the R-intersection of a polyhedral function and an ellipsoidal one, we can compute the R-function associated to the polyhedral function Fx (3) by following the procedure (5)-(6) of the previous subsection. 674

Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Here we denote the composed R-function with R p, where the subscript p indicates the R-intersection associated to the polyhedral function. Then we define the R-function associated to the quadratic function x Px R q (x) = x Px. (9) Finally the R-intersection is computed in accordance to the composition rule of Table as (x) = Rp(x)+Rq(x) R p(x) 2 +R q(x) 2 2αR p(x)r q(x) 2 2 2α () and the candidate CLF is Vα (x) = (x). () Remark 4. If both R p (for all hyperplanes intersections) and the final R-intersection () are computed with α =, then the standard truncated ellipsoid (8) is obtained. In other words, the truncated ellipsoid is a special case of the smoothing performed with the R-functions framework. 3.2 Lyapunov-based control synthesis A function V is a suitable CLF if the condition x R n : V(x)B = V(x)Ax } =, (2) is satisfied, as stated in [Petersen and Barmish (987b)]. In the above equations, V(x) is the gradient of function V(x) and is a row vector of zeros of appropriate dimensions. In practice, condition (2) implies that whenever the control action is ineffective, function V should decrease just the same, in fact the time derivative of the CLF V is V(x(t),u(t)) = V(x(t))Ax(t)+ V(x(t))Bu(t). (3) If condition (2) is satisfied, then there exists a control law u(t) such that V(x(t),u(t)) is always negative. The control law u(t) that minimizes V(x(t),u(t)), over the set U, is the (sliding) control u(x(t)) = sign ( B V(x(t)) ) (4) so that the time derivative of V becomes m V(x(t)) = V(x(t))Ax(t) ( V(x)B) i, (5) i= where ( V(x)B) i is the i th component of the row vector V(x)B. Condition (2) is not sufficient to guarantee that the time derivative (5) is always negative, because here the control u (4) is constrained in U. To overcome this feasibility problem, it is possible to derive a Petersen-like condition that guarantees that V is a suitable CLF by means of a constrained state feedback control ( u(t) ): } m x X : V(x)Ax ( V(x)B) i =. (6) i= A possible way of checking equation (6) is by solving an optimization problem. For instance, if it is required that the state should be constrained inside the intersection of a polyhedron represented by matrix F and an ellipsoid represented by the positive definite matrix P, the optimization problem } m max V(x)Ax ( V(x)B) x X i s.t. i= (7) X = x R n : Fx, x Px } has to be solved. In particular, condition (6) is satisfied if and only if the solution of (7) is negative. In this work we solveproblem(7)toprovethecorrectnessoftheproposed smoothed CLF. The drawback of the control law (4) is that it is highly discontinuous over time and often not implementable on real actuators. The discontinuity caused by the sign function can be avoided by approximating the control law (4) with arbitrary precision by using u(x(t)) = sat ( κb V(x(t)) ), (8) for κ R + sufficiently large [Blanchini (29)], where sat is the component-wise vector saturation function. It is particularly convenient to associate a gradient-based control to an everywhere differentiable CLF, because the corresponding control law is continuous over time [Petersen and Barmish (987a)]. 4. SIMULATIONS 4. Comparison with smoothed polyhedral functions In this subsection, we consider the examples proposed in [Blanchini and Miani (998)], where the control law is a gradient-basedcontrolassociatedtoaplf,smoothedwith standard 2p-norms. The framework of R-functions is used to smooth the inner sublevel sets of the polyhedral function Fx. The candidate CLFs are computed by following the procedure (5)-(6) with α = and they have been proved to be suitable CLFs for the corresponding constrained control problem by solving (7). Example concerns the constrained stabilization of the the dynamical system ) (ẋ (t) = [ ]( ) [ ]( ) x (t) u (t) +. (9) ẋ 2 (t) 4 (t) u 2 (t) The CLF proposed in [Blanchini and Miani (998)] is the polyhedral function Fx 2p, where [ ].5.5 F =, (2).5.5 with p = 3. The controlled region is Fx, that it is larger than the one ( Fx 2p ) provided in [Blanchini and Miani (998)], because also the corners of the polyhedral region are included. In the table, values of typical control indices are shown: ISE is the Integral of the Squared Error values and it should be small to avoid large state errors; ISTE is the Integral Square Time Error and it also should be small to avoid large state errors or slow convergence; T represents thetime ofconvergence (2-norm ofthe statevector smaller 6742

Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Table 2. Average control performances, normalized with respect to the results of [Blanchini and Miani (998)]. Results have been obtained averaging over simulations starting from random initial states inside the intersection region. Example Fx 2p R-function (α = ).2832.823.298.948 Example 2 Fx 2p R-function (α = ).947.5997.6557.762.8.6.4.2.2.4.6.8.8.6.4.2.2.4.6.8 x Fig. 3. Controlled state trajectories converging to the origin according to the R-function CLF. than 3 ); finally IADU is the Integral of the Absolute value of the time Derivative of the control signal u and it is desired to be small to avoid stress of the control actuator. With the use of R-functions all performance indices are improved, sincethenon-homothetic sublevelsetsprovidea smoother state convergence with respect to high-order 2pnorms. Normalized simulation results, averaged over simulations, are shown in Table 2. Example 2 [Blanchini and Miani (998)] concerns the constrained stabilization of the dynamical system characterized by matrices A =, B =. (2) Thestateconstraintis x andasuitablepolyhedral CLF Fx with..6.328 F =.25.762.225...25.6.762.983.26..25.762.656.26 (22) Also in this examples, the use of R-functions yields better control performances. The comparison is shown in Table 2. Some controlled state trajectories, starting from random initial points are shown in Figure 3. 4.2 Comparison with truncated ellipsoids Examples 3 and 4 are taken from [O Dell and Misawa (22)]. For each example, the candidate CLF Vα is Table 3. Average control performances, normalized with respect to the results of [Thibodeau et al. (29)] (Example 3) and [O Dell and Misawa (22)] (Example 4). Results have been obtained averaging over simulations starting from random initial states inside the intersection region. Example 3 Truncated Ellipsoid R-function (α = ).9478.99.956.9928 Example 4 Truncated Ellipsoid R-function (α = ).59.988.963.8533 computed by following the procedure (5)-(6) with α = to compute R p, together with the procedure (9)-() again with α =. Finally Vα has been proved to be a suitable CLF for the constrained control problem by solving problem (7). Example 3. The constrained stabilization of the dynamical system characterized by matrices [ ] [ ] A = 2, B =, F = I 3, (23) has been also addressed in [Thibodeau et al. (29)]. The controlsynthesizedin[o DellandMisawa(22)]islinear, i.e. u = Kx, where K = [.36.53.67]. (24) In [Thibodeau et al. (29)] the same control of [O Dell and Misawa (22)] is used, as solving the optimization procedure to find both optimal P and K matrices would require the solution of a BMI problem. Example 4 [O Dell and Misawa (22)]. The dynamical system is the double integrator [ ] [ ẋ(t) = x(t)+ u(t), (25) ] with state constraints x 25, 5 and control constraint u. A truncated ellipsoid and the static controller are synthesized in [O Dell and Misawa (22)]. Such static state feedback control is compared with the gradient-based control associated to the truncated ellipsoid smoothed with R-functions. The gradient-based control is smoother and it yields much faster convergence. Moreover, the use of a static state feedback control yields an undesirable oscillating behavior of the state trajectory, as it is shown in Figure 4. Finally, Table 4 shows that the computational time required by the nonlinear control law (8) associated to the smooth R-function CLF is actually comparable with the case in which (8) is associated to usual CLFs. The linear state feedback control is not considered in the comparison as it is obviously faster. 6743

Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 8 6 4 2 2 4 6 8 8 6 4 2 2 4 6 8 25 2 5 5 5 5 2 25 x 25 2 5 5 5 5 2 25 x Fig. 4. Controlled state trajectories converging to the origin according to the truncated ellipsoid CLF together with a static state feedback controller (top) and according to the R-function CLF together with the use of a gradient-based controller (bottom). Table 4. Average required computational time of equation (8), normalized with respect to the use of a PCLF. Results have been obtained averaging over state points. V V 2p V te Vα Example.39.5.72 Example 2.59.85.52 Example 3.48.63.95 Example 4.39.54.74 5. CONCLUSION In this paper a solution for the constrained stabilization problem of linear systems has been proposed. The control law minimizes the time derivative of a smoothed control Lyapunovfunctionwithinthesetofboundedcontrols.The novelty of the approach follows from the reinterpretation of polyhedral functions and truncated ellipsoids in terms of R-functions, providing a general framework to smooth both polyhedral and truncated ellipsoidal functions in a new non-homothetic way, which it has been shown to be convenient in terms of control performances. A Petersen-like condition has been proposed to check if a smoothed Lyapunov function candidate is a suitable control Lyapunov function by means of a gradient-based bounded control. The main contribution is the use of a gradient-based control together with the introduced smoothed truncated ellipsoidal Lyapunov function for constrained stabilization of linear systems, since the use of everywhere differentiable functions allow the control to be continuous over time. Although only quadratic and polyhedral functions have been investigated in this paper, the proposed framework allows for a flexible composition of arbitrary positive definite functions. Future work will also focus on casting the proposed Petersen-like condition into an easily-tractable LMI problem. REFERENCES Balestrino, A., Caiti, A., Crisostomi, E., and Grammatico, S. (29a). R-composition of Lyapunov functions. Proc. of the IEEE Mediterranean Conference on Control and Automation, Thessaloniki (Greece). Balestrino, A., Caiti, A., Crisostomi, E., and Grammatico, S. (29b). Stability analysis of dynamical systems via R-functions. Proc. of the IEEE European Control Conference, Budapest (Hungary). Balestrino, A., Caiti, A., Crisostomi, E., and Grammatico, S. (2). Stabilizability of linear differential inclusions via R-functions. Proc. of the IFAC Nonlinear Control Systems Conference, Bologna (Italy). Blanchini, F. (995). Nonquadratic Lyapunov functions for robust control. Automatica, 3(3), 45 46. Blanchini, F. (29). Lyapunov methods in robustness. An introduction. Lecture notes in Automatic Control, Bertinoro (Italy). Blanchini, F. and Miani, S. (998). Constrained stabilization via smooth Lyapunov functions. Systems & Control Letters, 35, 55 63. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (994). Linear matrix inequalities for the control of uncertain linear systems. Society for Industrial and Applied Mathematics (SIAM). Brayton, R. and Tong, C. (98). Constructive stability and asymptotic stability of dynamical systems. IEEE Transactions on Circuits and Systems, 27(), 2 3. Gutman, P. and Cwikel, M. (986). Admissible sets and feedback control for discrete-time linear systems with bounded control and states. IEEE Transactions on Automatic Control, 6, 373 376. O Dell, B. and Misawa, E. (22). Semi-ellipsoidal controlled invariant sets for constrained linear systems. Journal of Dynamic Systems, Measurement and Control, 24, 98 3. Petersen, I.R. and Barmish, B.R. (987a). The continuity of the minimum effort control law. Proc. of the IEEE Conference on Decision and Control, Los Angeles (California, USA). Petersen, I.R. and Barmish, B.R. (987b). Control effort considerations in the stabilization of uncertain dynamical systems. Systems & Control Letters, 9, 47 422. Polanski, A. (2). On absolute stability analysis by polyhedrallyapunovfunctions. Automatica,36(4),573 578. Rvachev, V. (982). Theory of R-functions and some applications (in Russian). Naukova Dumka. Shapiro, V. (27). Semi-analytic geometry with R- functions. ACTA numerica, 6, 239 33. Thibodeau, T., Tong, W., and Hu, T. (29). Set invariance and performance analysis of linear systems via truncated ellipsoids. Automatica, 45, 246 25. 6744