INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust. Nonlinear Control 2011; 21:2079–2100 Published online 30 December 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1682
Generalized nonlinear H∞ synthesis condition with its numerically efficient solution Qian Zheng1 and Fen Wu2, ∗, † 1 Department
of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, U.S.A. of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, U.S.A.
2 Department
SUMMARY In this paper, we will first derive a general synthesis condition for the output-feedback H∞ control of smooth nonlinear systems. Computationally efficient H∞ control design procedure for a subclass of smooth nonlinear systems with polynomial vector field is then proposed by converting the resulting Hamilton-Jacobi-Isaacs inequalities from rational forms to their equivalent polynomial forms. Using quadratic Lyapunov functions, both the state-feedback and output-feedback problems will be reformulated as semi-definite optimization conditions and locally tractable solutions can be obtained through sum-ofsquares (SOS) programming. The proposed nonlinear H∞ design approach achieves significant relaxations on the plant structure compared with existing results in the literature. Moreover, the SOS-based solution algorithm provides an effective computational scheme to break the bottleneck in solving nonlinear H∞ and optimal control problems. The proposed nonlinear H∞ control approach has been applied to several examples to demonstrate its advantages over existing nonlinear control techniques and its usefulness to engineering problems. Copyright 䉷 2010 John Wiley & Sons, Ltd. Received 22 May 2009; Revised 13 October 2010; Accepted 27 October 2010 KEY WORDS:
nonlinear H∞ control; output feedback; polynomial nonlinear systems; SOS programming
1. INTRODUCTION The analysis and control of nonlinear systems are among the most challenging problems in systems and control theory. In the past decade, a theoretical framework for exploring H∞ control of nonlinear systems has been proposed in [1–3]. Interpreting nonlinear H∞ control in terms of dissipativity and differential game [4], the solution to this problem has been related to an appropriate Hamilton-Jacobi-Isaacs (HJI) equation. For hyperbolic nonlinear systems whose linearized plant are stabilizable, the solution of H∞ state-feedback control was characterized [1, 3] by an invariant manifold of Hamiltonian vector fields using differential geometric theory. Later, the result has been generalized to non-hyperbolic nonlinear systems via output-feedback control [2]. It was further shown that the solution to output-feedback control problem is determined by a pair of coupled HJI equations. Parallel to linear H∞ control theory, a separation principle was also established under a detectability hypothesis [5]. However, solving nonlinear H∞ control problems in a numerically efficient way and making it practical to real-world problems remains an unsolved issue. It is well known that the HJI partial differential equation (PDE) reduces to Riccati algebraic equations for linear systems, which can be solved easily by efficient numerical algorithms. In the ∗ Correspondence
to: Fen Wu, Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, U.S.A. † E-mail:
[email protected] Copyright 䉷 2010 John Wiley & Sons, Ltd.
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nonlinear context, however, there is no systematic numerical algorithm currently available for the solution of this PDE. Therefore, the key of nonlinear H∞ control theory is the solvability of HJI equations. To this end, various approaches have been proposed to solve the HJI equation numerically. In [6, 7], Taylor series expansions of the storage function V (x) are considered to solve the HJI equation term by term in an iterative fashion, provided that the linearized model of the nonlinear system has a solution. However, the solutions from these approaches do not have a closed form and they may not converge to an analytic solution. In [8], the HJI solution was also determined by solving the coefficient of finite series expansion of the value function. Nevertheless, it is difficult to guarantee stability of the closed-loop system using finite truncation of the series. Moreover, the stability region is limited by the convergence domain of the underlying Taylor series, which is not available a priori. Alternatively, successive approximations reduce the HJI equation to an infinite sequence of linear partial differential equations; the solution of which are then obtained by the Galerkin method [9] or neural networks [10]. In these schemes, finite truncation of the algorithm would result in stabilizing control laws arbitrarily close to the true solution. However, the region of convergence is usually not defined explicitly and these approximate methods only consider state feedback control problems. This is a significant drawback since in most engineering applications, it would be either too expensive to have sensors for each state, or very difficult to measure some states. On the other hand, a convex parameterization of nonlinear output feedback H∞ control condition is derived in [11] based on a pair of positive definite matrix functions P(x), Q(x). Unfortunately, it is difficult, if not impossible, to specify the functional form of P(x), such that *V (x)/*x = 2x T P(x) except for the trivial case when P(x) is a constant matrix. Therefore, the proposed synthesis condition does not naturally lead to computationally tractable solution algorithms for nonlinear H∞ control. In [12], the L2 gain analysis problem for polynomial nonlinear systems is formulated as a convex state-dependent linear matrix inequality (LMI), which can be recast as a sum-of-squares (SOS) optimization problem. This approach is promising for overcoming the numerical difficulty in solving HJI inequality and provides an analytic solution at the same time. SOS programming has also been applied to solve other nonlinear analysis [13, 14] and stabilization [15, 16] problems. The major advantages of SOS decomposition are the resulting computational tractability and the algorithmic character of its solution procedure [17]. This could help to provide a coherent methodology of synthesizing Lyapunov functions for nonlinear systems. In addition, the importance of SOS technique also lies in its ability to provide tractable relaxations for many difficult optimization problems including nonlinear H∞ and optimal controls. The difficulties associated with the nonlinear H∞ control problem have motivated research work on alternative nonlinear control approaches which avoid the unwieldy task of solving the HJI equation directly [18, 19]. Using backstepping control techniques, robust control Lyapunov function (RCLF) can be constructed manually for nonlinear systems. The nonlinear stabilizing control law derived from such a Lyapunov function guarantees that the closed-loop system is robustly stable against uncertainties and disturbances in a bounded set [18, 20]. The RCLF-based design is often useful in establishing global stability, but it falls short in optimizing control performance for nonlinear systems. The latest development of backstepping techniques for nonlinear control includes inverse optimal control [18, 21] and local optimal control designs [22]. The idea of inverse optimal control is to construct a Lyapunov function first, then design a control law which justifies that this control strategy is optimal with respect to some objective function. Although the resulting controller often leads to satisfactory results, the cost function is determined a posteriori and it may not meet the requirement of control designers. On the other hand, the local optimal H∞ control problem has been addressed in [22]. By combining Taylor series expansion and backstepping design, global stabilizing control laws are constructed to match the optimal controller at the origin up to a desired order. Nevertheless, the optimality only holds locally around the origin. After all, the backstepping control approach is limited to pure feedback (lower-triangular) systems and the design procedure is often cumbersome with recursive construction of stabilizing Lyapunov functions. In this paper, we will first extend the HJI inequalities, which are instrumental for nonlinear H∞ synthesis, to their most general forms by relaxing orthonormal and decoupling assumptions and by including the non-zero direct feedthrough term D11 into the nonlinear plant model. Moreover, Copyright 䉷 2010 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control 2011; 21:2079–2100 DOI: 10.1002/rnc
SOLUTION TO GENERALIZED NONLINEAR H∞ CONTROL
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we will focus on materializing the generalized H∞ control theory into an algorithmic procedure for a class of nonlinear systems with polynomial vector field. It turns out that the generalized H∞ synthesis conditions are composed of rational matrix inequalities and can be converted into polynomial forms in a local region. The nonlinear H∞ control problem will then be formulated as semi-definite (convex) optimization conditions based on the idea of SOS decomposition. More specifically, we will use quadratic Lyapunov functions to convert original HJI inequalities into state-dependent LMIs for polynomial nonlinear systems. Consequently, testing non-negativity of the generalized Gramian matrix from the resulting LMIs is solvable using SOS programming techniques with polynomial-time complexity. An academic nonlinear mass-spring-damper problem will be used to demonstrate the nonlinear H∞ design procedure. Other than this example, the proposed approach is also applied to a practical nonlinear spacecraft control problem expressed in the polynomial form. Note that any smooth nonlinear dynamics can be approximated by polynomial functions on a compact set. The applicability of this nonlinear control design approach has also been broadened by the fact [23] that many non-polynomial systems including aircraft [24] and automotive engines can be converted to their polynomial models through algebraic transformations. A preliminary version of this paper was published in [25] to solve H∞ control problem for polynomial nonlinear systems. It was shown that higher order Lyapunov functions are helpful to improve controlled performance. Nevertheless, the state-feedback synthesis condition in [25] was non-convex polynomial matrix inequalities and had to be solved by an iterative algorithm. Moreover, without considering direct feedthrough term D11 , the synthesis condition becomes simpler and easy to handle. The notation used in this paper is fairly standard, and will be defined when it is necessary. We denote non-negative integer set as Z+ . R stands for the set of real numbers and R+ for the non-negative real numbers. Rm×n is the set of real m ×n matrices. We use Sn×n to denote real, symmetric n ×n matrices, and Sn×n for positive-definite matrices. A block diagonal matrix + with matrices X 1 , X 2 , . . . , X p on its main diagonal is denoted by diag{X 1 , X 2 , . . . , X p }. In large symmetric matrix expressions, terms denoted by ‘’ are to be induced by symmetry. The notation Fu (M, ) stands for an upper LFT, which is defined as M11 M12 Fu , = M22 + M21 (I − M11 )−1 M12 . M21 M22 A lower LFT F (M, ) will be defined similarly. For two integers k1 and k2 , k1 0, we define T T = −D1122 − D1121 (2 I − D1111 D1111 )−1 D1111 D1112
B1 (x) = B1 (x)+ B2 (x)D21 (x)
= A(x)+ B2 (x)C2 (x) A(x) 1 (x) = C1 (x)+ D12 (x)C2 (x) C
11 (x) = D11 + D12 (x)D21 (x). D
11 (x)]0, we have T T 1 T 1 T T Tˆ ˆ v v −r r dt eˆ eˆ − d d dt. 2 2 0 0 Therefore, if the transformed system (17) is asymptotically stable in the region X and has its H∞ norm less than from r to v, so does the system (6) for which the H∞ norm is defined from d to e. The relationship between the two systems is depicted in Figure 1. (II) Output estimation (OE) synthesis condition for the system with simpler structure For the system (17) with OE structure, it was proven in [26] that a controller in the form of T x˙c = Aa (xc )xc + Ba (xc )[ Dˆ 12 (x) Dˆ 12 (x)] 2 F0 (xc )+ L 0 (xc )[Ca (xc )xc − y] 1
T u¯ = [ Dˆ 12 (x) Dˆ 12 (x)] 2 F0 (xc ) 1
(18)
T (x)]−1 will render the closedwith L 0 (x) satisfying (*V (x)/*x)L 0 (x) = −2x T CaT (x)[ Dˆ 21 (x) Dˆ 21 loop system asymptotically stable and has its H∞ norm less than in a local region within X if
Copyright 䉷 2010 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control 2011; 21:2079–2100 DOI: 10.1002/rnc
SOLUTION TO GENERALIZED NONLINEAR H∞ CONTROL
w
z
G
r
2087
v
GT
y
K
y
K ||v||2 ||r||2
||z||2 0 find Si s.t. −v2T NFC (Si , i , Ri−1 )CFC (x)v2 ∈ SOS 2 * −1 v3 ∈ SOS −v3T [N (S , , R )C (x)] FC i FC i i 2 x=0 *x x T (Si −i−2 Ri−1 )x ∈ SOS
(25)
where v2 , v3 are free variables of compatible dimension. 5. If i is feasible, continue; otherwise, go back to step 1. 6. If i−1 −i 0, CFC (x)>0 validate the synthesis condition resulting from the plant transformation. Therefore, the obtained solution is generally non-unique and will depend on how the designer incorporates the above constraints into the controller synthesis condition. Comparing with the standard nonlinear H∞ counterpart, the polynomial forms of D12 , D21 will lead to multiplicative terms CFI (x), CFC (x) for the generalized synthesis condition. As a result, the relaxed D11 , D12 (x), D21 (x) does increase the computational cost in the proposed solution procedure. The computational complexity will be determined by the nonlinear plant size, and the resulting SOS polynomial degrees. The number of optimization variables include that of actual decision variables, variables used for parameterization of regional multipliers, and other variables due to coupling constraints among monomials. For a polynomial of degree 2d in n o variables (n o is the number of states plus that of auxiliary variables), the maximal dimension of monomial vector n o +d for SOS decomposition will be . Depending on the structure of the nonlinear dynamics as d well as the implementation of SOS solver, the actual dimension of the monomial vector for SOS decomposition could be much less than its maximum bound. Table I provides some representative dimensions of monomial vectors obtained using our developed monomial reduction algorithm versus the maximal dimensions of monomial vectors. Superscripts ‘*’ are the numbers obtained in solving some nonlinear H∞ control problems and others are based on random examples run by our algorithm. It is clear to see that our monomial reduction algorithm is quite efficient and scales well with the problem size.
4. EXAMPLES In this section, the proposed nonlinear H∞ control design procedure will be illustrated on an academic example and compared with the existing nonlinear control approach. Moreover, its applicability to solve practical engineering problems will be demonstrated through a nonlinear spacecraft example. 4.1. Nonlinear mass-spring-damper system First, we will apply the proposed nonlinear output-feedback H∞ control design procedure to a nonlinear mass-spring-damper system. The dynamic equation of the nonlinear system is given by m x¨ + g(x, x)+ ˙ f (x) = (x)u Copyright 䉷 2010 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control 2011; 21:2079–2100 DOI: 10.1002/rnc
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where m is the mass, u is the control force, f (x) represents a nonlinear spring term, g(x, x) ˙ is the damping term, and (x) is the nonlinearity associated with control input. The parameters are chosen to be m = 1kg
g(x, x) ˙ = c1 x˙
f (x) = c2 x +c3 x 3
(x) = 1+c4 x 2
c1 = 1 kg/s c2 = −0.01 kg/s2
c3 = 0.1 kg s/m2
c4 = 0.13/m2 .
Note that the nonlinear spring provides negative linear spring force for small displacement, but presents positive force for large x. There are two types of disturbances acting on the system. One is external input such as aerial force affecting the dynamics of the mass velocity. The other is measurement noise acting on the output channel. Define x1 = x (displacement) and x2 = x˙ (velocity). By incorporating disturbances and defining controlled and measured outputs, the second-order mechanical system will be written in polynomial form as 0 1 0 x˙1 x1 0 0.8 d1 = + + u (26) 0.8 0 x˙2 d2 0.01−0.1x12 −1 x2 1+0.13x12 e1 0.6 0 x1 0.5 0 d1 0 = + + u (27) 0 0.3 x2 0 0.5 d2 0.5 e2 x1 d1 y = [1 1] +[0 1+0.5x1 ] . (28) x2 d2 Choosing an initial 0 = 2.5 and applying Lemma 1, one can transform the nonlinear massspring-damper system to its equivalent system with Dˆ 11 = 0 and ⎤ ⎡ 556+92x1 −25x12 −16+8x1 ⎥ ⎢ 108+20x1 −5x12 5(108+20x1 −5x12 ) ⎥ ⎢ ⎥ ⎢ ˆ = A(x) ⎥ ⎢ 3 3 2 4 2 4 ⎣129.88+22.20x1 −24.57x1 −3.3x1 +0.5x1 −164−30x1 +0.72x1 −1.3x1 +0.39x1⎦ 108+20x1 −5x12
⎡ ⎢ ⎢ Bˆ 1 (x) = ⎢ ⎢ ⎣
(108+20x1 −5x12 )
1.3064
⎡
(108+20x1 −5x12 )
Dˆ 21 (x) = 0
⎤
0
⎥ ⎦
0.5657 (108+20x1 −5x12 )
5.6569(2+ x1 ) 1
(108+20x1 −5x12 ) 2
⎤
25(108+20x1 −5x12 )
1
1 2
2− x1 16(108+20x1 −5x12 )
⎢ ⎥ ⎢ ⎥ Bˆ 2 (x) = ⎢ ⎥ ⎣ 2200+500x1 +286x12 +65x13 ⎦
35.3553(108+20x1 −5x12 ) 2
−2.8284
⎡
⎥ ⎥ ⎥ ⎥ ⎦
1 2
−(100+13x12 )(2+ x1 )
0.9798
⎢ Cˆ 1 (x) = ⎣
⎤
9.051
0
108+20x1 −5x12
⎡ ⎢ Dˆ 12 (x) = ⎣
1 2
⎤ 88+20x1 ⎢ 108+20x1 −5x12 ⎥ ⎥ ⎢ Cˆ 2 (x) = ⎢ ⎥ ⎣ 2(56+10x1 −3x12 ) ⎦ ⎡
0
108+20x1 −5x12 ⎤ ⎥ ⎦.
5.6569 1
(108+20x1 −5x12 ) 2
The transformed system is well defined when CFI (x):=108+20x1 −5x12 >0 or −3.0596
