CLF-based Nonlinear Control with Polytopic Input Constraints
Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
WeM07-4
CLF-based Nonlinear Control with Polytopic Input Constraints J. Willard Curtis Air Force Research Laboratory Eglin AFB, FL Abstract This paper presents a practical method for generating high-performance control laws which are guaranteed to be stabilizing in the presence of known input constraints. We address the class of smooth nonlinear systems which are affine in the control with non-smooth (rectangular or polytopic) actuator constraints and a known control Lyapunov function. We present a result which uses a complete state-dependent description of the stabilizing control value set to generate, point-wise, the set of input values which contains all the (Lyapunov) stabilizing control values that simultaneously obey the input constraints. The vertex enumeration algorithm is then used to derive a complete parameterization of this set, and a nonlinear program is employed to select a high-performance control from this feasible and stabilizing control set for an illustrative example.
1
Introduction
Ensuring good performance, via state-dependent feedback control, for the class of nonlinear systems is a challenging problem. Some effective methods for the design of such controllers are linearization [9, p. 485 – 488], gain scheduling [11], feedback linearization [8], recursive integrator backstepping [15], statedependent Riccati control [4], and Lyapunov-based methods. Strategies based on Lyapunov functions are particularly valuable because they offer analytical guarantees such as global asymptotic stability, inverseoptimality, and robustness to uncertainty [7]. Most Lyapunov-based methods for nonlinear systems can interpreted in a control Lyapunov function [16, 1] (clf) framework. Thus, clfs have become a powerful tool for the feedback stabilization of nonlinear systems even though finding a method of finding
0-7803-7924-1/03/$17.00 ©2003 IEEE
a clf for a general nonlinear system remains an open problem (though there are such clf-building techniques for systems in cascade form [10, 15] and systems which are amenable to feedback linearization). There exist several ‘universal formulas’ that, given a known clf, generate almost smooth control laws which are guaranteed to globally asymptotically stabilize the closedloop system such as Sontag’s formula [17] and Freeman and Kokotovic’s point-wise min-norm formula [7]. An important extension to the theory of clf-based control is emerging for nonlinear systems that are subject to input constraints [12, 13, 14, 20, 19]. This is a deserving area of research due to the fact that the vast majority of real feedback systems have saturating actuators, and the design of stabilizing controllers for constrained systems is a long standing problem even for linear systems (see [18] for some recent contributions). In [13] the authors introduce a universal formula for the case of a scalar input which is constrained to be positive and possibly bounded, and in [14] a clfbased formula is derived for inputs that lie within a Minkowski ball. An interesting technique is presented in [20] where it is shown how control Lyapunov functions can be used in a receding horizon formulation to stabilize systems with input constraints, but this receding-horizon approach is computationally intensive and may not be amenable to a real-time implementation. Another promising result is found in [19] where the authors construct a one-parameterized family of universal formulas for systems with very general input constraints. This paper extends those constrained-clf results by presenting a complete parameterization of all universal formulas that stabilize a nonlinear system under polytopic input constraints with respect to a known clf. In particular we build upon the approach presented in [5, 6] where it is shown that Lyapunov stability can be interpreted as a point-wise one-dimensional
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constraint on control values and where a complete parameterization of unconstrained universal formulas is developed. We show that the point-wise stability requirement, V˙ < 0, can be interpreted as adding another bounding hyperplane to the input constraint polytope. From this half-space representation we exploit the vertex enumeration algorithm introduced in [3, 2] to find a vertex representation of the polytope. It is then shown how a vector of weights can be used to completely parameterize the set of feasible stabilizing control values at any given state. The benefit of this parameterization is that it can be combined with an optimization routine to improve closed-loop performance, while not requiring the intense computational load associated with recedinghorizon control. Thus, the technique in this paper represents a practical method for constructing highperformance stabilizing controllers for nonlinear affine systems with polytopic input constraints.
2
Control Lyapunov Functions
Consider the following affine nonlinear system with multiple inputs, i.e., x˙ = f (x) + g(x)u,
(1)
2−D Control Space 5
4 T
Case where Vx f = 2. 3
2
1
0 −gTVx −1
−2
−3
This shaded region contains all control values that render Lf+guV negative.
−4
−5 −5
−4
−3
−2
−1
0
1
2
3
4
5
Figure 1: Stability Constraint along the ith semi-axis. (In other words U is a hyperbox which encloses the origin and whose faces are perpendicular to the axes. The ith face of U lies a distance b(i) from the origin.) Definition 2.2 a C 1 function V (x) : Rn → R is said to be a control Lyapunov function (clf ) for system (1) if V (x) is positive definite, radially unbounded, and if inf VxT (f + gu) < 0,
u∈U
for all x 6= 0.
where x ∈ R , u ∈ U ⊂ R and f (0) = 0. We will assume throughout the paper that f and g are locally Lipschitz functions, and the goal will be to regulate the state x to the origin.
We assume that a clf for the constrained system is known a priori. Also, it is assumed that there is a minimum desired rate of decrease, V˙ = VxT (f + gu) ≤ −²(kxk), where ²(kxk) is a positive-definite function and Vx (x) is the gradient of V with respect to x; this assumption will ensure a closed control value set.
Definition 2.1 A polytope, P ⊂ Rm , is the bounded intersection of a finite number of closed halfspaces:
2.1
n
4
©
m
m
P = z∈R
T
ª
: z hi ≤ bi , 2m ≤ i ≤ k < ∞
We assume that the control values are constrained to lie in some polytope, U, which contains the origin. This input constraint can be written compactly as follows: U = {u ∈ Rm : M u ≤ b} , (2) where the matrix M ∈ Rk×m and the vector b ∈ Rk define the polytope. For the ubiquitous special case when U is a hyper-box, M ∈ R2m×m can be represented as · ¸ 4 I M= , −I and the vector b is of length 2m with the ith element of b containing the magnitude of the control constraint
The Stabilization Constraint
Definition 2.3 The Stabilizing Set, denoted S(x) is a state-dependent control value set containing all the points in Rm which satisfy: ª 4 © S(x) = u ∈ U: VxT (f + gu) ≤ −²(kxk) , (3) Definition 2.2 guarantees that S(x) is always nonempty if ²(kxk) is chosen to be sufficiently small. S is thus a closed state-dependent half-space in rem . Definition 2.4 Given a constraint polytope, U ⊂ Rm , A control law u(t) is feasible if u(·) ∈ L∞ and u(t) ∈ U for all t ≥ t0 . It can be trivially shown that any continuous selection from S will be a feasible control and will render the closed loop system asymptotically stable (since V˙ < 0 ∀ x 6= 0 and u ∈ S =⇒ u ∈ U).
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constraint, as outlined in Theorem 2.5, will render one or more of the input constraints redundant or that the stability constraint will itself be redundant. For example, it is possible that any control value in U will stabilize the system. In this case, the new system of inequalities described by (4) has an inactive constraint because the last row of M u ≤ b (VxT gu < −Vx T f − ²(kxk)) is redundant.
2−D Control Space 5
4
3
2
1
0
T
−g Vx −1 This triangle is the feasible and stabilizing set of control values: S
−2
−3
Other than needlessly enlarging the size of the inequality constraints, this redundancy can impair our parameterization and optimization algorithms. Fortunately, we can remove this redundancy with a simple linear program. To accomplish this consider the following system of inequalities:
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−5 −5
−4
−3
−2
−1
0
1
2
3
4
5
Figure 2: Stabilizing Polytope The next theorem shows how the stability constraint embodied in (3) can be folded into the input constraints described by (2). © ª Theorem 2.5 S(x) = u ∈ Rm : M u < b , where M and b are defined as follows: ¸ · M , M= VxT g ¸ · b . b= −VxT f − ²(kxk) Proof:
sT ξ ≤ t. We can determine whether sT ξ ≤ t is redundant by performing the following linear program: f ∗ = max sT ξ, subject to Aξ ≤ d sT ξ ≤ t + m kbk∞ .
The matrix inequality Mu ≤ b
(4)
represents a system of inequalities. The first k rows ensure that u ∈ U. The last (k + 1) row, VxT gu < −Vx T f − ²(kxk) is simply a restatement of VxT f + VxT gu < −²(kxk). ¥ Note that Theorem (2.5) reveals an important result: asymptotic stability at any fixed state can be viewed as a linear constraint on the input (see Figure 1). This reduces the design of a feasible, stabilizing control to performing a (continuous in the state) point-wise selection from the well-defined set S(x). Figure 2 shows a two dimensional example of the the state-dependent set S at some state where VxT f + ²(kxk) = 2, VxT g = (1 1)T , and U is rectangular. Note that S is always closed, always convex, and always a polytope.
3
Aξ ≤ d,
Removing Redundancy and Vertex Enumeration
Before delving into a parameterization of S, we note that it is possible that the addition of the stability
Then sT ξ ≤ t is redundant if f ∗ ≤ t. Each row in (4) is sequentially tested and removed if redundant. This process is repeated until no redundancy is found. In order to put the constraints in a form amenable to a convenient parameterization, it will first be necessary to find the vertices of the convex polytope defined by (4). This problem can be solved via the vertex enumeration algorithm introduced in [3]. We will not repeat the details of this algorithm, but its function is explained below. The Minkowski-Weyl Theorem [21, p. 29] states that every polytope P can be described as the intersection of a finite set of half spaces, P = {ξ ∈ Rm : Aξ ≤ d}, called the H − representation, or described equivalently in terms of the convex hull of its vertices V = {v1 , v2 , ...vN }, vi ∈ Rm , and N < ∞, termed its V − representation. The vertex enumeration algorithm transforms a polytope’s H − representation into its V − representation. More specifically, given (4) which has been pruned of its redundant inequalities, the algorithm generates a set of points V = {v1 , v2 , ...vN }, vi ∈ Rm corresponding to the (exterior) vertices of the polytope S ⊂ Rm .
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4
Parameterizing S
4
The constrained stabilizing control value set, S(x), can be completely parameterized in terms of its vertices. Definition 4.1 Given a polyhedron’s V − representation, we define the m × N vertex matrix Vm and the N × 1 weighting vector w as follows: £ ¤ Vm = v1 v2 · · · vN , £ ¤ wT = w1 w2 · · · wN , PN where i=1 wi = 1 and wi ≥ 0.
Max-Rate Control
The control law which is developed in [19] is defined point-wise as the control value which lies (almost on) the boundary of the constraint set. Furthermore, this value is chosen such that the maximum (point-wise) rate of decrease in V is achieved. Given our notation and assumptions, their control law would be written as: uM R = max −V˙ . (6) Under our parameterization, this universal formula can be solved via a simple linear program: maximize −VxT gV w over w at every state.
∀ u ∈ S ∃ w such that u = V w, where w and V are defined by Definition 4.1. Proof: Follows directly from the fact that the S is a closed, convex polytope and from the fact that V w is a convex weighting of points on the surface of S. ¥ The weighting vector w therefore defines a particular convex combination of the vertices, resulting in a point on the interior or on the surface of S. Thus, if we know the V − representation of S at some state, then S is completely parameterized by the weighting vector w at that state. We have now reduced the problem of designing a stabilizing controller in the presence of input constraints to the point-wise choice of an N −dimensional weighting vector. Since we will now explore optimization routines for choosing the weighting vector, a remark on the size of N is in order: when the input constraints are rectangular, the polytope S will have 2m faces and 2m vertices.
5.1
5.2
u∈S
Theorem 4.2
5
where ² = αV . Under our parameterization, the MNF can be easily implemented via a quadratic program, by choosing the weighting vector such that kuk = wT V T V w is minimized at every state.
Optimizing the Weighting Vector Min-norm Control
Freeman and Kokotovic’s min-norm formula (MNF) is based on the reasonable notion that control effort is a valuable resource. In [7] the MNF is defined point-wise as the control value of minimum norm that produces a desired rate of decrease (αV ) in the clf. Given the our assumptions and notation, the MNF control value at every state can be described as uM N F = min kuk , u∈S
(5)
5.3
Receding Horizon Control
Implementing a receding-horizon strategy similar to the one proposed in [20] can now be easily accomplished. First, we discretize the time index (xk ≈ x(kδt)), then we define an Euler approximation to the system dynamics, and we define a discrete cost function J(u, x): x0 = x(0),
¡ ¢ xk = xk−1 + f (xk−1 ) + g(xk−1 )V (xk−1 )w(i) · δt, J(u, xk ) =
k+T X
xTk Qxk ,
k
where T is some integer that controls the length of the objective function horizon. The optimization problem can now be posed as follows. Find a sequence of weighting vectors ¡ ¢ w(0), w(1), · · · , w(i), · · · , w(imax ) such that J is minimized. The number of time steps for which each w(i) is held constant, the length of the time horizon, the number of distinct weighting vectors (imax ) and the size of the time step are all parameters that can be tuned in order to trade off accuracy and computational load. Having found such an optimal sequence, the first element, wk , is used as our control law, and the optimization is repeated at regular intervals. This is the standard receding horizon formulation, but note that this is an any-time algorithm: the optimization can be halted at any point and the current weighting vector used to build the control law. Also, any valid weighting vector can be used to initialize the optimization.
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Figure 3: State Trajectories under u1
Figure 4: State Trajectories under u2 20
6
Illustrative Example
15
10
In order to demonstrate how this strategy can be applied in practice, we illustrate the design of a control law for a two-dimensional unstable linear system:
5
0
−5
x˙ = Ax + Bu,
(7) −10
2
with the input, u(x) ∈ R , constrained to lie in a rectangle, Ω, containing the origin:
−15
−20 −20
4
©
2
−15
−10
−5
0
5
10
15
20
ª
Ω = ξ ∈ R : − 20 ≤ ξ(1) ≤ 15, −15 ≤ ξ(1) ≤ 10 . (8) We choose the matrices A and B such that the unconstrained system is open-loop unstable and controllable: · ¸ · ¸ 4 4 1 0 1 −.2 A= ,B = . −.1 1 0 1 We will use V (x) = xT P x, where P is the solution to the associated Riccati equation, for the clf, and we note that this will serve as a clf only when the system’s initial conditions satisfy some bound, kxk < `(|P |, Ω). (For the following plots the initial state was x0 = (−12, 12)T ). We tested and compared two control laws in simulation. u1 = V (x)w1 was implemented with an equal weight on each vertex of S. u2 = V (x)w2 was implemented by weighting equally the two vertices furthest from the origin to create a form of max-norm control. Figures 3 and 4 show the state trajectories under these selection strategies. Note that u2 regulated the state to the origin significantly faster than u1 , though at the expense of larger control effort. This illustrates that the weighting vector is an effective way to tune a control in order to achieve performance objectives. For illustrative purposes Figures 5 and 6 show the sets S for u1 and u2 respectively, and the small circle represents the control
Figure 5: S(x) and u1 (x) value selected by the corresponding weights. (Note that these are snapshots taken at different states.)
7
Discussion and Conclusion
This paper introduced a method for algorithmically parameterizing stabilizing control laws that obey polytopic input constraints, given a known clf. This technique is a general method, being appropriate for the class of smooth nonlinear systems which are affine in the control. Our approach relies on a fast (polynomial time) algorithm to generate the vertices of a statedependent polytope, and it is amenable to real-time implementation. In particular we have demonstrated the following: • Lyapunov stability is equivalent to a point-wise inequality constraint on the input, and this constraint is expressed in a form where it can be easily folded into rectangular or polytopic input constraints. • The set of simultaneously feasible and stabilizing controls is a polytope in Rm that can be com-
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[6] Jess W. Curtis and Randal W. Beard. A graphical understanding of lyapunov-based nonlinear control. In Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV, December 2002.
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[7] R. A. Freeman and P. V. Kokotovic. Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Systems and Control: Foundations and Applications. Birkhauser, 1996.
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[8] Alberto Isidori. Nonlinear Control Systems. Communication and Control Engineering. Springer Verlag, New York, New York, second edition, 1989.
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[9] Hassan K. Khalil. Nonlinear Systems. Macmillan Publishing Company, New York, New York, 1996.
Figure 6: S(x) and u2 pletely parameterized by a weighting of its vertices. • Any universal formula can be represented via our parameterization. We have also shown that powerful convex optimization techniques can be applied to create a point-wise selection from this polytope, and we show that such selections can produce large gains in performance for an illustrative, open-loop unstable, example problem. In general, the importance of selecting an appropriate control value from the feasible and stabilizing set depends on the ‘size’ of this set in the control space. In problems where this control value set is large, especially when the dimension of the control space is large, correspondingly large gains in performance can be expected when an optimization is used to dynamically select the vertex weighting vector.
References [1] Z. Artstein. Stabilization with relaxed controls. Nonlinear Analysis, Theory, Methods, and Applications, 7(11):1163–1173, 1983. [2] David Avis, David Bremner, and Raimund Seidel. How good are convex hull algorithms. Computational Geometry, 7:265–301, 1997. [3] David Avis and Komei Fukudo. Reverse search for enumeration. Discrete Applied Mathematics, 65:21– 46, 1996. [4] James R. Cloutier. Adaptive matched augmentation proportional navigation. In Proceedings of the AIAA Missile Sciences Conferenence, Monterey, California, Nov 1994. [5] Jess W. Curtis and Randal W. Beard. Satisficing: A new approach to constructive nonlinear control. IEEE Transactions on Automatic Control, submitted 2001.
[10] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic. Nonlinear and Adaptive Control Design. Wiley, 1995. [11] D. A. Lawrence and W. J. Rugh. Gain scheduling dynamic linear controllers for a nonlinear plant. Automatica, 31:380–390, 1995. [12] Y. Lin and E. Sontag. A universal formula for stabilization with bounded controls. Systems Control Letters, 16:393–397, 1991. [13] Y. Lin and E. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control-Theory and Advanced Technology, 10:1981–2004, December 1995. [14] M. Malisoff and E. Sontag. Universal formulas for CLF’s with respect to Minkowski balls. In Proceedings of the European Control Conference, Brussels, Belgium, July 1997. [15] R. Sepulchre, M. Jankovic, and P. Kokotovic. Constructive Nonlinear Control. Communication and Control Engineering Series. Springer-Verlag, 1997. [16] E. D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM Journal on Control and Optimization, 21:462–471, May 1983. [17] E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 34:435–443, 1989. [18] A. Stoorvogel and A. Saberi. Editors, Special issue on control of systems with bounded control. Int. J. Robust and Nonlinear Control, 9, 1999. [19] Rodolfo Suarez, Julio Solis-Daun, and Baltazar Aguirre. Global clf stabilization for systems with compact convex control value sets. In Proceedings of the IEEE Conference on Decision and Control, Orlando, FL, December 2001. [20] Mario Sznaier and Rodolfo Suarez. Suboptimal control of constrained nonlinear systems via receding horizon state dependent Riccati equations. In Proceedings of the IEEE Conference on Decision and Control, Orlando, FL, December 2001. [21] G. M. Ziegler. Lectures on Polytopes. Springer, 1996.
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CLF-based Nonlinear Control with Polytopic Input Constraints J. Willard Curtis Air Force Research Laboratory Eglin AFB, FL Abstract This paper presents a practical method for generating high-performance control laws which are guaranteed to be stabilizing in the presence of known input constraints. We address the class of smooth nonlinear systems which are affine in the control with non-smooth (rectangular or polytopic) actuator constraints and a known control Lyapunov function. We present a result which uses a complete state-dependent description of the stabilizing control value set to generate, point-wise, the set of input values which contains all the (Lyapunov) stabilizing control values that simultaneously obey the input constraints. The vertex enumeration algorithm is then used to derive a complete parameterization of this set, and a nonlinear program is employed to select a high-performance control from this feasible and stabilizing control set for an illustrative example.
1
Introduction
Ensuring good performance, via state-dependent feedback control, for the class of nonlinear systems is a challenging problem. Some effective methods for the design of such controllers are linearization [9, p. 485 – 488], gain scheduling [11], feedback linearization [8], recursive integrator backstepping [15], statedependent Riccati control [4], and Lyapunov-based methods. Strategies based on Lyapunov functions are particularly valuable because they offer analytical guarantees such as global asymptotic stability, inverseoptimality, and robustness to uncertainty [7]. Most Lyapunov-based methods for nonlinear systems can interpreted in a control Lyapunov function [16, 1] (clf) framework. Thus, clfs have become a powerful tool for the feedback stabilization of nonlinear systems even though finding a method of finding
0-7803-7924-1/03/$17.00 ©2003 IEEE
a clf for a general nonlinear system remains an open problem (though there are such clf-building techniques for systems in cascade form [10, 15] and systems which are amenable to feedback linearization). There exist several ‘universal formulas’ that, given a known clf, generate almost smooth control laws which are guaranteed to globally asymptotically stabilize the closedloop system such as Sontag’s formula [17] and Freeman and Kokotovic’s point-wise min-norm formula [7]. An important extension to the theory of clf-based control is emerging for nonlinear systems that are subject to input constraints [12, 13, 14, 20, 19]. This is a deserving area of research due to the fact that the vast majority of real feedback systems have saturating actuators, and the design of stabilizing controllers for constrained systems is a long standing problem even for linear systems (see [18] for some recent contributions). In [13] the authors introduce a universal formula for the case of a scalar input which is constrained to be positive and possibly bounded, and in [14] a clfbased formula is derived for inputs that lie within a Minkowski ball. An interesting technique is presented in [20] where it is shown how control Lyapunov functions can be used in a receding horizon formulation to stabilize systems with input constraints, but this receding-horizon approach is computationally intensive and may not be amenable to a real-time implementation. Another promising result is found in [19] where the authors construct a one-parameterized family of universal formulas for systems with very general input constraints. This paper extends those constrained-clf results by presenting a complete parameterization of all universal formulas that stabilize a nonlinear system under polytopic input constraints with respect to a known clf. In particular we build upon the approach presented in [5, 6] where it is shown that Lyapunov stability can be interpreted as a point-wise one-dimensional
2228
constraint on control values and where a complete parameterization of unconstrained universal formulas is developed. We show that the point-wise stability requirement, V˙ < 0, can be interpreted as adding another bounding hyperplane to the input constraint polytope. From this half-space representation we exploit the vertex enumeration algorithm introduced in [3, 2] to find a vertex representation of the polytope. It is then shown how a vector of weights can be used to completely parameterize the set of feasible stabilizing control values at any given state. The benefit of this parameterization is that it can be combined with an optimization routine to improve closed-loop performance, while not requiring the intense computational load associated with recedinghorizon control. Thus, the technique in this paper represents a practical method for constructing highperformance stabilizing controllers for nonlinear affine systems with polytopic input constraints.
2
Control Lyapunov Functions
Consider the following affine nonlinear system with multiple inputs, i.e., x˙ = f (x) + g(x)u,
(1)
2−D Control Space 5
4 T
Case where Vx f = 2. 3
2
1
0 −gTVx −1
−2
−3
This shaded region contains all control values that render Lf+guV negative.
−4
−5 −5
−4
−3
−2
−1
0
1
2
3
4
5
Figure 1: Stability Constraint along the ith semi-axis. (In other words U is a hyperbox which encloses the origin and whose faces are perpendicular to the axes. The ith face of U lies a distance b(i) from the origin.) Definition 2.2 a C 1 function V (x) : Rn → R is said to be a control Lyapunov function (clf ) for system (1) if V (x) is positive definite, radially unbounded, and if inf VxT (f + gu) < 0,
u∈U
for all x 6= 0.
where x ∈ R , u ∈ U ⊂ R and f (0) = 0. We will assume throughout the paper that f and g are locally Lipschitz functions, and the goal will be to regulate the state x to the origin.
We assume that a clf for the constrained system is known a priori. Also, it is assumed that there is a minimum desired rate of decrease, V˙ = VxT (f + gu) ≤ −²(kxk), where ²(kxk) is a positive-definite function and Vx (x) is the gradient of V with respect to x; this assumption will ensure a closed control value set.
Definition 2.1 A polytope, P ⊂ Rm , is the bounded intersection of a finite number of closed halfspaces:
2.1
n
4
©
m
m
P = z∈R
T
ª
: z hi ≤ bi , 2m ≤ i ≤ k < ∞
We assume that the control values are constrained to lie in some polytope, U, which contains the origin. This input constraint can be written compactly as follows: U = {u ∈ Rm : M u ≤ b} , (2) where the matrix M ∈ Rk×m and the vector b ∈ Rk define the polytope. For the ubiquitous special case when U is a hyper-box, M ∈ R2m×m can be represented as · ¸ 4 I M= , −I and the vector b is of length 2m with the ith element of b containing the magnitude of the control constraint
The Stabilization Constraint
Definition 2.3 The Stabilizing Set, denoted S(x) is a state-dependent control value set containing all the points in Rm which satisfy: ª 4 © S(x) = u ∈ U: VxT (f + gu) ≤ −²(kxk) , (3) Definition 2.2 guarantees that S(x) is always nonempty if ²(kxk) is chosen to be sufficiently small. S is thus a closed state-dependent half-space in rem . Definition 2.4 Given a constraint polytope, U ⊂ Rm , A control law u(t) is feasible if u(·) ∈ L∞ and u(t) ∈ U for all t ≥ t0 . It can be trivially shown that any continuous selection from S will be a feasible control and will render the closed loop system asymptotically stable (since V˙ < 0 ∀ x 6= 0 and u ∈ S =⇒ u ∈ U).
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constraint, as outlined in Theorem 2.5, will render one or more of the input constraints redundant or that the stability constraint will itself be redundant. For example, it is possible that any control value in U will stabilize the system. In this case, the new system of inequalities described by (4) has an inactive constraint because the last row of M u ≤ b (VxT gu < −Vx T f − ²(kxk)) is redundant.
2−D Control Space 5
4
3
2
1
0
T
−g Vx −1 This triangle is the feasible and stabilizing set of control values: S
−2
−3
Other than needlessly enlarging the size of the inequality constraints, this redundancy can impair our parameterization and optimization algorithms. Fortunately, we can remove this redundancy with a simple linear program. To accomplish this consider the following system of inequalities:
−4
−5 −5
−4
−3
−2
−1
0
1
2
3
4
5
Figure 2: Stabilizing Polytope The next theorem shows how the stability constraint embodied in (3) can be folded into the input constraints described by (2). © ª Theorem 2.5 S(x) = u ∈ Rm : M u < b , where M and b are defined as follows: ¸ · M , M= VxT g ¸ · b . b= −VxT f − ²(kxk) Proof:
sT ξ ≤ t. We can determine whether sT ξ ≤ t is redundant by performing the following linear program: f ∗ = max sT ξ, subject to Aξ ≤ d sT ξ ≤ t + m kbk∞ .
The matrix inequality Mu ≤ b
(4)
represents a system of inequalities. The first k rows ensure that u ∈ U. The last (k + 1) row, VxT gu < −Vx T f − ²(kxk) is simply a restatement of VxT f + VxT gu < −²(kxk). ¥ Note that Theorem (2.5) reveals an important result: asymptotic stability at any fixed state can be viewed as a linear constraint on the input (see Figure 1). This reduces the design of a feasible, stabilizing control to performing a (continuous in the state) point-wise selection from the well-defined set S(x). Figure 2 shows a two dimensional example of the the state-dependent set S at some state where VxT f + ²(kxk) = 2, VxT g = (1 1)T , and U is rectangular. Note that S is always closed, always convex, and always a polytope.
3
Aξ ≤ d,
Removing Redundancy and Vertex Enumeration
Before delving into a parameterization of S, we note that it is possible that the addition of the stability
Then sT ξ ≤ t is redundant if f ∗ ≤ t. Each row in (4) is sequentially tested and removed if redundant. This process is repeated until no redundancy is found. In order to put the constraints in a form amenable to a convenient parameterization, it will first be necessary to find the vertices of the convex polytope defined by (4). This problem can be solved via the vertex enumeration algorithm introduced in [3]. We will not repeat the details of this algorithm, but its function is explained below. The Minkowski-Weyl Theorem [21, p. 29] states that every polytope P can be described as the intersection of a finite set of half spaces, P = {ξ ∈ Rm : Aξ ≤ d}, called the H − representation, or described equivalently in terms of the convex hull of its vertices V = {v1 , v2 , ...vN }, vi ∈ Rm , and N < ∞, termed its V − representation. The vertex enumeration algorithm transforms a polytope’s H − representation into its V − representation. More specifically, given (4) which has been pruned of its redundant inequalities, the algorithm generates a set of points V = {v1 , v2 , ...vN }, vi ∈ Rm corresponding to the (exterior) vertices of the polytope S ⊂ Rm .
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4
Parameterizing S
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The constrained stabilizing control value set, S(x), can be completely parameterized in terms of its vertices. Definition 4.1 Given a polyhedron’s V − representation, we define the m × N vertex matrix Vm and the N × 1 weighting vector w as follows: £ ¤ Vm = v1 v2 · · · vN , £ ¤ wT = w1 w2 · · · wN , PN where i=1 wi = 1 and wi ≥ 0.
Max-Rate Control
The control law which is developed in [19] is defined point-wise as the control value which lies (almost on) the boundary of the constraint set. Furthermore, this value is chosen such that the maximum (point-wise) rate of decrease in V is achieved. Given our notation and assumptions, their control law would be written as: uM R = max −V˙ . (6) Under our parameterization, this universal formula can be solved via a simple linear program: maximize −VxT gV w over w at every state.
∀ u ∈ S ∃ w such that u = V w, where w and V are defined by Definition 4.1. Proof: Follows directly from the fact that the S is a closed, convex polytope and from the fact that V w is a convex weighting of points on the surface of S. ¥ The weighting vector w therefore defines a particular convex combination of the vertices, resulting in a point on the interior or on the surface of S. Thus, if we know the V − representation of S at some state, then S is completely parameterized by the weighting vector w at that state. We have now reduced the problem of designing a stabilizing controller in the presence of input constraints to the point-wise choice of an N −dimensional weighting vector. Since we will now explore optimization routines for choosing the weighting vector, a remark on the size of N is in order: when the input constraints are rectangular, the polytope S will have 2m faces and 2m vertices.
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u∈S
Theorem 4.2
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where ² = αV . Under our parameterization, the MNF can be easily implemented via a quadratic program, by choosing the weighting vector such that kuk = wT V T V w is minimized at every state.
Optimizing the Weighting Vector Min-norm Control
Freeman and Kokotovic’s min-norm formula (MNF) is based on the reasonable notion that control effort is a valuable resource. In [7] the MNF is defined point-wise as the control value of minimum norm that produces a desired rate of decrease (αV ) in the clf. Given the our assumptions and notation, the MNF control value at every state can be described as uM N F = min kuk , u∈S
(5)
5.3
Receding Horizon Control
Implementing a receding-horizon strategy similar to the one proposed in [20] can now be easily accomplished. First, we discretize the time index (xk ≈ x(kδt)), then we define an Euler approximation to the system dynamics, and we define a discrete cost function J(u, x): x0 = x(0),
¡ ¢ xk = xk−1 + f (xk−1 ) + g(xk−1 )V (xk−1 )w(i) · δt, J(u, xk ) =
k+T X
xTk Qxk ,
k
where T is some integer that controls the length of the objective function horizon. The optimization problem can now be posed as follows. Find a sequence of weighting vectors ¡ ¢ w(0), w(1), · · · , w(i), · · · , w(imax ) such that J is minimized. The number of time steps for which each w(i) is held constant, the length of the time horizon, the number of distinct weighting vectors (imax ) and the size of the time step are all parameters that can be tuned in order to trade off accuracy and computational load. Having found such an optimal sequence, the first element, wk , is used as our control law, and the optimization is repeated at regular intervals. This is the standard receding horizon formulation, but note that this is an any-time algorithm: the optimization can be halted at any point and the current weighting vector used to build the control law. Also, any valid weighting vector can be used to initialize the optimization.
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Figure 3: State Trajectories under u1
Figure 4: State Trajectories under u2 20
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Illustrative Example
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In order to demonstrate how this strategy can be applied in practice, we illustrate the design of a control law for a two-dimensional unstable linear system:
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with the input, u(x) ∈ R , constrained to lie in a rectangle, Ω, containing the origin:
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Ω = ξ ∈ R : − 20 ≤ ξ(1) ≤ 15, −15 ≤ ξ(1) ≤ 10 . (8) We choose the matrices A and B such that the unconstrained system is open-loop unstable and controllable: · ¸ · ¸ 4 4 1 0 1 −.2 A= ,B = . −.1 1 0 1 We will use V (x) = xT P x, where P is the solution to the associated Riccati equation, for the clf, and we note that this will serve as a clf only when the system’s initial conditions satisfy some bound, kxk < `(|P |, Ω). (For the following plots the initial state was x0 = (−12, 12)T ). We tested and compared two control laws in simulation. u1 = V (x)w1 was implemented with an equal weight on each vertex of S. u2 = V (x)w2 was implemented by weighting equally the two vertices furthest from the origin to create a form of max-norm control. Figures 3 and 4 show the state trajectories under these selection strategies. Note that u2 regulated the state to the origin significantly faster than u1 , though at the expense of larger control effort. This illustrates that the weighting vector is an effective way to tune a control in order to achieve performance objectives. For illustrative purposes Figures 5 and 6 show the sets S for u1 and u2 respectively, and the small circle represents the control
Figure 5: S(x) and u1 (x) value selected by the corresponding weights. (Note that these are snapshots taken at different states.)
7
Discussion and Conclusion
This paper introduced a method for algorithmically parameterizing stabilizing control laws that obey polytopic input constraints, given a known clf. This technique is a general method, being appropriate for the class of smooth nonlinear systems which are affine in the control. Our approach relies on a fast (polynomial time) algorithm to generate the vertices of a statedependent polytope, and it is amenable to real-time implementation. In particular we have demonstrated the following: • Lyapunov stability is equivalent to a point-wise inequality constraint on the input, and this constraint is expressed in a form where it can be easily folded into rectangular or polytopic input constraints. • The set of simultaneously feasible and stabilizing controls is a polytope in Rm that can be com-
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[6] Jess W. Curtis and Randal W. Beard. A graphical understanding of lyapunov-based nonlinear control. In Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV, December 2002.
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[7] R. A. Freeman and P. V. Kokotovic. Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Systems and Control: Foundations and Applications. Birkhauser, 1996.
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[8] Alberto Isidori. Nonlinear Control Systems. Communication and Control Engineering. Springer Verlag, New York, New York, second edition, 1989.
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[9] Hassan K. Khalil. Nonlinear Systems. Macmillan Publishing Company, New York, New York, 1996.
Figure 6: S(x) and u2 pletely parameterized by a weighting of its vertices. • Any universal formula can be represented via our parameterization. We have also shown that powerful convex optimization techniques can be applied to create a point-wise selection from this polytope, and we show that such selections can produce large gains in performance for an illustrative, open-loop unstable, example problem. In general, the importance of selecting an appropriate control value from the feasible and stabilizing set depends on the ‘size’ of this set in the control space. In problems where this control value set is large, especially when the dimension of the control space is large, correspondingly large gains in performance can be expected when an optimization is used to dynamically select the vertex weighting vector.
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