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A practical method for the design of sliding mode controllers using linear matrix inequalities
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Automatica 40 (2004) 1761 – 1769
www.elsevier.com/locate/automatica
Brief paper
A practical method for the design of sliding mode controllers using linear matrix inequalities Christopher Edwards∗ Control and Instrumentation Group, Department of Engineering, University of Leicester, LE1 7RH, UK Received 18 September 2003; received in revised form 14 February 2004; accepted 10 May 2004
Abstract This paper considers the problem of selecting a sliding surface for a given system in an optimal way such that the performance of the reduced order system is balanced against the control costs required to maintain sliding. This approach is quite di0erent from many of the existing schemes in the literature in which no direct account is taken of the control e0ort. Two new numerical schemes are presented both involving convex optimization problems. Three simple examples are given to demonstrate the theory. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Sliding modes; Quadratic cost functions; LMIs
1. Introduction Sliding mode control is an established method of controlling uncertain dynamical systems (Utkin, 1992; Edwards & Spurgeon, 1998). Its invariance properties with respect to so-called matched uncertainty has encouraged researchers to apply sliding mode techniques to a wide variety of application areas (Utkin, 1992; Edwards & Spurgeon, 1998; Utkin, Guldner, & Shi, 1999; Hung, Gao, & Hung, 1993). The early theory was developed within a state-space framework and invariably assumed that full state information was available for use in the control law. The design of a (state feedback) sliding mode controller traditionally involves,
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. ∗ Tel.: +44-116-223-1303; fax: +44-116-252-2619. E-mail addresses: [email protected], [email protected] (C. Edwards). 0005-1098/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.05.004
Linear Matrix Inequality (LMI) methods have been explored (Arzelier, Angulo, & Bernussou, 1997; Choi, 1997, 1999; Herrmann, Spurgeon, & Edwards, 2001). However, from the standpoint of designing a control law for real systems, with limits on the available control action, all these methods have shortcomings. Because of the two-stage nature of the design process, at the time of synthesizing the switching function, no account is taken of the control action that is necessary to induce, and more importantly maintain sliding, even for nominal linear systems. For instance, the optimal quadratic method of Utkin and Young (1978) which resembles the traditional linear quadratic control methods, deliberately has zero penalty on the control e0ort and adopts the principle of ‘cheap control’ (Utkin & Young, 1978; Utkin, 1992). This paper seeks to develop a framework to design the switching function whilst quantifying the control action that is necessary to maintain sliding. This requirement will be formulated as a convex optimization problem consisting of coupled Linear Matrix Inequalities. Linear Matrix Inequalities have been used previously for sliding mode control law design (Arzelier et al., 1997; Choi, 1997, 1999; Herrmann et al., 2001). In Arzelier et al. (1997) LMIs have been used to place the poles of the reduced order sliding motion in convex regions of the complex plane. The emphasis in the work of Choi (1997, 1999) has been largely the design of sliding surfaces so that the sliding motion is robust to unmatched uncertainty. In all these methods no
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C. Edwards / Automatica 40 (2004) 1761 – 1769
explicit account is taken of the resulting control signals required to induce and maintain sliding. In the recent design method of Herrmann et al., 2001, linear quadratic methods are employed and a two-stage optimization is performed. However in terms of the hyperplane design, essentially the method of Utkin and Young (1978) is used (although LMI methods are employed to obtain the solution) and hence no penalties are imposed in terms of the control. This paper is organized as follows: the next section describes the problem that will be addressed in this paper; Section 3 considers two LMI-based methods to synthesize the required sliding surface; Section 4 considers three different simple examples to demonstrate the theory;
(SB)−1 s(t) governs the (asymptotic) rate of convergence to the sliding surface when (·) = 0. This follows since substituting the control law (2) into the nominal linear system (1), the switching function satis
(5)
(1)
(assuming (·) = 0) and thus s(t) → 0 asymptotically. In what follows, as in Choi (1997), it will be assumed that = Im where is a negative scalar. Consequently P2 can be any s.p.d matrix. Without loss of generality it will be assumed that P2 = Im . The control law in (2)–(3) is now entirely parameterized by S and . In designing the sliding surface, i.e. selecting the matrix S, it is usually convenient to assume the system is in regular form (Utkin, 1992). Even though for the method which will be proposed in this paper a di0erent coordinate system will be proposed, as a starting point, it will be assumed that (by a change of coordinates) the system matrices in (1) have the form A11 A12 0 A= ; B= ; (6) Im A21 A22
where the state x ∈ R and the control signal u ∈ R . The unknown signal : R+ ×Rn ×Rm → Rm represents matched uncertainty and is assumed to be norm bounded by a known function (t; x; u). Assume the input distribution matrix B is full rank and the pair (A; B) is controllable. The objective is to design a switching surface of the form
where A11 ∈ R(n−m)×(n−m) . The structure of the input distribution matrix above is more stringent than that normally imposed for regular form, but is necessary for the exposition which follows, and can be achieved very easily. 1 In this coordinate system the switching function matrix may be parameterized so that
S = {x : Sx = 0};
S = S2 [M Im ];
2. Problem statement Consider the uncertain linear system x(t) ˙ = Ax(t) + B(u(t) + (t; x; u)); n
m
where S ∈ Rm×n is a full rank matrix which needs to be designed so that the associated reduced order sliding mode, when the system states are con
(2)
m×m
is a stable matrix and P2 s(t) un (t) = − (t; x; u)(SB)−1 if s(t) = 0; P2 s(t)
(3)
where P2 ∈ Rm×m is a symmetric positive de
(4)
The scalar function (·) must be chosen as an appropriate upper bound on the matched uncertainty satisfying (t; x; u) ¿ SB(t; x; u). Remark 2.1. This is a commonly adopted control structure in the literature (Ryan & Corless, 1984; Zinober, 1994; Edwards & Spurgeon, 1998). The linear part of the control law in (2) can be thought of as comprising a term −(SB)−1 SAx(t) which represents the so-called ‘equivalent control’ necessary to maintain a sliding motion on S (Utkin, 1992) in the absence of uncertainty, whilst the term
m×(n−m)
(7) m×m
where M ∈ R and S2 ∈ R is nonsingular. Theoretically the choice of S2 has no e0ect on the reduced order sliding motion. If (x1 ; x2 ) represents a partition of the states associated with the canonical form in (6), then, from (7), during the sliding motion, s = 0 and so x2 = −Mx1 . Substituting for x2 in the
(8)
and so the sliding motion is thus governed by the system matrix (A11 − A12 M ). If there is no uncertainty in (1) and letting (·) → 0 the control law in (2) becomes linear. The objective is to design the switching function matrix M in (7) so that the control e0ort arising from the nominal linear control law u(t) = −(SB)−1 (SA − S)x(t)
(9)
minimizes a given cost function. This approach of ignoring the matched uncertainty and designing the sliding surface based on the nominal linear model is quite consistent with existing switching function design methods (Utkin, 1992; Zinober, 1994). 1
From a numerical viewpoint it is often convenient to use an orthogonal change of coordinates to achieve regular form (Zinober, 1994; Edwards & Spurgeon, 1998). However in order to obtain the canonical form in (6), a non-orthogonal coordinate change would normally be required.
C. Edwards / Automatica 40 (2004) 1761 – 1769
A plausible formulation is therefore to attempt to choose S and so that the closed-loop system resulting from using the linear control law (9) minimizes the cost functional ∞ x()T Qx() + u()T Ru() d; (10) J=
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and hence ˜ = (A11 − A12 M ) ∪ {}m : (A˜ + B˜ L)
where Q ∈ Rn×n is a given symmetric semi-de
This partition of the eigenvalues and the repeated appearance of is expected from the structure of the controller in (9) and the choice of =Im . Indeed this partition of the closed-loop poles and the eigenstructure associated with (5) is inherent in the approaches described in Ryan and Corless (1984); Zinober (1994), Edwards and Spurgeon (1998). The canonical form in (12)–(13) will now be shown to be a useful one in which to synthesize the design freedom encapsulated by M . Two di0erent approaches for the selection of M will now be given in the following subsections.
L = [M Im ](In − A):
3.1. A min–max approach
0
(11)
The parameter M thus represents the design freedom in the problem. Remark 2.2. Although this paper assumes that all the states are available for use in the controller, the scheme could still be implemented if only measured outputs were available, by the use of an observer. As shown in (Edwards & Spurgeon, 1996, 1997), if the nominal system triple relating inputs to measured outputs is relative degree one and minimum phase, by using an appropriate sliding mode observer, all the robustness properties of state feedback sliding mode control laws with respect to matched uncertainty can be retained.
as
Observe that the gain matrix L˜ from (14) can be expressed
L˜ = M [In−m 0(n−m)×m ] + [0 Im ]:
(17)
From this view point the closed-loop system matrix can be written as 0 A11 −A12 A˜ + B˜ L˜ = + MC2 ; (18) A21 Im Im − A22 B2
Ap
where C2 = [In−m 0(n−m)×m ]
3. LMI formulations Now consider the change of coordinates x → (In − A)x. Because, by design, does not belong to the spectrum of A, this is a nonsingular transformation. It will of course (generally) destroy the structure in the input distribution matrix traditionally associated with regular form in (6). In ˜ B) ˜ has the form this new coordinate system the new pair (A; A A 11 12 A˜ = (In − A)A(In − A)−1 = A = ; (12) A21 A22 −A12 B˜ = (In − A)B = : (13) Im − A22 The linear state feedback gain in (11) takes the form L˜ = [M Im ](In − A)(In − A)−1 = [M Im ]
(14)
and the switching function from (7) becomes S˜ = S2 [M Im ](In − A)−1 : It can be readily veri
0 Im
and the design problem becomes a static output feedback LQ problem involving the
(16)
(20)
where B1 = (In − A), w(t) is an exogenous disturbance and the control law is given by u˜ = MC2 x. ˜ The optimization problem of minimizing J from (10) can be formulated as an H2 problem. Let Qq ∈ Rq×n with q 6 n be a full row rank matrix such that QqT Qq = Q from (10). Then formally the equivalent H2 problem can be posed as: Minimize with respect to M the H2 norm of the error signal ˜ z = C1 x + D12 u; where
Qq; 1
C1 : = 0 0
(15)
(19)
(21) Qq; 2
R1=2 0m×m
0q×m
D12 : = 0m×m
(22)
R1=2
and [Qq; 1 Qq; 2 ] := Qq (In − A)−1 . The appearance of the R1=2 term in the C1 matrix comes from the
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C. Edwards / Automatica 40 (2004) 1761 – 1769
function matrix S and control gain L can be obtained from Eqs. (7) and (11). 3.2. A block diagonal approximation As an alternative to the min–max scheme (which requires an iterative solution), a di0erent approach will be considered in this subsection. Again the coordinates associated with (12)–(13) will be used. The minimization of the cost function (10) can be cast as an optimization problem in terms of LMIs, namely: Minimize trace((In − A)T X −1 (In − A)) subject to ˜ + X (A + B˜ L) ˜ T XQqT X L˜T R1=2 (A + B˜ L)X Qq X −Iq 0 ¡ 0; 1=2 ˜ 0 −Im R LX (23) X ¿0
(24)
with respect to the decision variables L˜ and X where X ∈ Rn×n is a s.p.d matrix. Inequality (23) is (e0ectively) inequality 7.40 in Boyd, Ghaoui, Feron, and Balakrishnan (1994). The left and right multiplication of X −1 by (In −A) in the trace operation is required because of the change in coordinates from the special regular form, where the cost function in (10) is de
(n−m)×(n−m)
and X2 ∈ R
m×m
and de
˜ = [N1 X2 ]; N = LX
(26)
m×(n−m)
n×n
. Let Z ∈ R be a s.p.d. matrix of where N1 ∈ R ˜ from (26) in (23) ‘slack’ variables. Then substituting for LX yields the convex optimization problem: Minimize trace(Z) with respect to the variables X; Z and N subject to ˜ + (BN ˜ )T XQqT N T R1=2 AX + XAT + BN Qq X −Iq 0 ¡ 0; R1=2 N
−Z (In − A)
(In − A) −X
0 T
−Im (27)
¡ 0:
(28)
Standard LMI optimization algorithms such as those described in Gahinet, Nemirovski, Laub and Chilali (1995) can be used to obtain a numerical solution to this problem. Once X and N have been synthesized, the gain matrix L˜ can be recovered from the expression L˜ = NX −1 = [N1 X1−1 Im ]
(29)
and thus the design parameter which parameterizes the switching function M := N1 X1−1 : The switching function matrix S and control gain L can be obtained from Eqs. (7) and (11). Although the block diagonal structure enforced on X in (25) potentially introduces conservatism, this structure together with the structure of A˜ + B˜ L˜ in (16) does allow very neatly the introduction of additional constraints (with appropriate LMI representations) on the dynamics of the sliding motion in (8). Taking V (x1 )=x1T X1−1 x1 as a Lyapunov function for the states x1 it follows that pole-placement within standard convex regions of the complex plane such as circles, strips and cones can be incorporated as additional LMI constraints (Chilali and Gahinet, 1996; Gahinet et al., 1995) in terms of the variables X1 and N1 . The LMI representation of these regions will be aQne terms of X1 and N1 and so can be combined with (27)–(28) whilst still retaining a convex optimization problem. In addition, to guard against unmodelled un-matched uncertainties, an inequality of the form A11 X1 + X1 AT11 − A12 N1 − N1 AT12 + DDT + X1 E T EX1 ¡ 0
(30)
can be included where D and E are known matrices of appropriate dimension. This guarantees stability of the reduced order sliding motion in (8) against the e0ects of lumped parametric uncertainty of the form DREx1 (t) for all R ¡ 1. Remark 3.1. • One aspect of both approaches is that the canonical form —so called ‘regular form’—is not used as the basis for the design procedure. • In Herrmann et al. (2001) a two-stage optimization procedure was proposed based on H2 methods. In (Herrmann et al., 2001) the hyperplane (parameterized in terms of M ) was
C. Edwards / Automatica 40 (2004) 1761 – 1769
downside is that the scaled min–max algorithm is more computationally intensive because of its iterative nature 2 and has no convergence guarantees. A bene
(32)
and the state-feedback component of the controller L = [ − 9:8818 − 10:9882]:
(33)
The associated sliding motion is governed by a pole at −0:9882. The LMI computed minimal value for the trace((I2 − A)T X −1 (I2 − A)) with X block diagonal is 109.6819, with √ corresponding upper-bound on the associated H2 norm 109:6819 = 10:4729. With L as given in 2
The diagonalization approach requires the solution of 4n + q + m inequalities in n2 + n decision variables; whilst the min–max algorithm involves 4n + m + q + 5 inequalities in n2 + n + 2 decision variables.
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(33), the true value of the H2 cost can be computed from solving the Lyapunov equation Plq (A + BL) + (A + BL)T Plq = −QqT Qq − LT RL (34) trace(Plq ) = 8:4259. This gives an indifrom which cation of the conservatism introduced by enforcing a block-diagonal structure on X as in (25). For comparison a standard LQR calculation with these choices of Qq and R gives closed-loop poles at −2:2361 ± 2:2361i and a H2 cost of 7:0138. As argued earlier however this is not a feasible solution to the sliding mode controller problem. Using the scaled min–max algorithm (see Appendix) to solve the design problem formulated in Section 3.1 with an upper bound on the square of the H2 norm , = 169, and using initial conditions of I2 =100 and - = 1 the following results are obtained: Iteration
l
.
-
1 2 3 4
0.86448037 0.99680425 0.99685960 0.99684657
150.59570656 1.02019790 1.00065630 1.00117895
0.00574229 0.00562059 0.00560472 0.00558872
The fact that both l → 1 and . → 1 indicates the convergence of the algorithm (Iwasaki & Skelton, 1994). The associated Lyapunov matrix (which is the inverse of the one obtained from the min–max algorithm) 0:6619 −0:7123 X= −0:7123 20:3419 and the static output gain in (18) is given by M = −0:9876 which gives a switching function matrix S = [0:9876 1:0000] and state-feedback gain L = [ − 9:8758 − 10:9876]: The associated sliding motion is governed by a pole at −0:9876. The true value of the H2 cost in this case is 8:4274. This is almost identical to that given by the diagonalization method. It should be noted however that the solution obtained from the min–max algorithm depends (in this case) quite signi
C. Edwards / Automatica 40 (2004) 1761 – 1769
this is conservative: the actual cost obtained from solving (34) is 3.4510.) For comparison a standard LQR calculation with these choices of Qq and R gives closed-loop poles at {−10:0000; −2:0412} and an optimal value of 3:4467. The scaled min–max algorithm with , = 45 and initial conditions I2 =100 and - =1, after 10 iterations, gives a static output feedback gain matrix M = −1:3691. The switching function matrix
States
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L = [ − 13:6906 − 11:3691]: The actual H2 norm is 3:9849 and so in this case there is no improvement over the approximation method described in Section 3.2. Remark 4.1. Because in both case (a) and (b) the cost function matrix Q = QqT Qq is not positive de
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time, sec Fig. 1. Evolution of the states for = 0 and 1.
S = [1:3691 1:0000] and state-feedback gain
1 0.5 0 −0.5 −1 −1.5 −2 0
matrices are 0 1:0000 0 A= 0 0 −0:6000
0
88:7574 ; −24:0000
0 B= 0: 1
The states represent shaft position, shaft speed and a scaling of the current in the armature coils, respectively. 3 The objective here is to minimize the H2 cost problem described earlier, but also to ensure that the complex conjugate pair of poles governing the sliding motion have a damping ratio greater than 0:7660. In order to guarantee this, as argued in Section 3.2, the LMIs (27)–(28), can be augmented with constraints involving X1 and N1 . The sliding motion is governed by the system matrix (A11 − A12 M ) and so, as argued in Chilali and Gahinet (1996), Gahinet et al. (1995) if (A11 X1 + X1 AT11 − A12 N1 − N1T AT12 )sin/ (X1 AT11 − N1T AT12 − A11 X1 + A12 N1 )cos/ (A11 X1 − A12 N1 − X1 AT11 + N1T AT12 )cos/ (A11 X1 + X1 AT11 − A12 N1 − N1T AT12 )sin/
¡0
then the eigenvalues of (A11 −A12 M ) lie in the sector centred at the origin making an angle of / with the negative real axis. To guarantee the damping ratio speci
This scaling is necessary to achieve the special case of regular form given in (6).
Switching function
C. Edwards / Automatica 40 (2004) 1761 – 1769 0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Control
Fig. 2. Evolution of the switching functions for = 0 and 1.
1 0 −1 −2 −3 −4 −5 −6
sary to establish the structure of the input distribution matrix in (6) yielding −0:0186 −0:0065 0:0064 −0:0012 0:0026 −0:1354 0:0020 −0:0028 A= ; −0:1311 0:0349 −0:4684 −0:0095 1:0120 0 0 0 0 B= ; 1 0
Time, sec
C= 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time, sec
Fig. 3. Evolution of the signals for = 0 and 1.
been performed with = 0 and 1. Fig. 1 is the response of the system regulating the angular position output from 1 rad back to zero. Very little di0erence is seen between the response of the linear and nonlinear controllers—particularly in terms of the position output. Fig. 2 shows the evolution of the switching functions. The
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−0:7236
−0:2741
0:6707
−0:1085
0:0043
0:0031
−0:2750
−0:1933
0:0030
0:0212
0
−0:1523
1
:
(35)
1 Choosing Qq = 10 C, R = I2 and = −0:5 the diagonalization procedure produced a sliding surface −0:0434 0:2080 1:0000 0:0000 S= −1:5478 1:0559 0:0000 1:0000
with the reduced order sliding motion governed by poles at {−0:0201; −0:1330}. For this controller, the associated H2 cost is 4:1069. For comparison, if the robust pole-placement algorithm in MATLAB’s Control toolbox is used to assign the reduced order poles at {−0:0201; −0:1330}, using the algorithm described in Section 4 in Edwards and Spurgeon (1998), (which could be described as a ‘traditional’ regular form based approach Utkin & Young, 1978)), the sliding surface matrix −0:0749 0:9924 −1:0000 −0:0000 S= : 0:8582 −0:1643 −0:0000 −1:0000 For the associated linear controller the H2 cost is 4:8829 which is signi
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C. Edwards / Automatica 40 (2004) 1761 – 1769
of the reduced order sliding mode; (c) despite some conservatism, very good results have been obtained on the examples considered. The second approach considers a more general problem but requires the use of the so-called scaled min–max algorithm which is much more computationally intensive. Three examples have been used to demonstrate the eQcacy of the method. Appendix A Consider the system x(t) ˙ = Ap x(t) + B1 w(t) + B2 u(t);
(A.1)
where w(t) is an exogenous disturbance and u(t) is the control input which is assumed to have the static output feedback form u = MC2 x, where M is the control gain to be designed. The objective is to select M so that the H2 norm of the error signal z = C1 x + D12 u
T PAcl + ATcl P + Ccl Ccl ¡ 0
(A.3)
and trace B1T PB1 ¡ ,;
(A.4)
where Acl = Ap + B2 MC2 and Ccl = C1 + D12 MC2 . The scaled min–max algorithm proposed by Iwasaki and Skelton (1994) to synthesize M depends on six variables: the scalars 2; -; .; l ∈ R and the s.p.d. matrices X; Y ∈ Rn×n . The iterative scaled min–max algorithm can be summarized as: Step 1: Choose initial values for Yopt ∈ Rn×n and -opt ¿ 0 and set k = 1. Step 2: Solve the following convex optimization problem: (a) Minimize .k with respect to X and 2 such that 1=2 1=2 In 6 Yopt XYopt 6 .k In
1 6 2-opt 6 .k ⊥ Ap X + XATp B2 C1 X
XC1T −2I
B2 D12
⊥T 6−I
to yield Xopt and 2opt . (b) With these values of Xopt and 2opt solve the following convex optimization problem: Maximize lk with respect to Y and - such that 1=2 1=2 lk In 6 Xopt YXopt 6I
lk 6 2opt - 6 1
C2⊥T (YAp + ATp Y + -C1T C1 )C2⊥
-
-
−I
trace B1T YB1 − ,-
-
-
−I
60
60
to yield Yopt and -opt . Step 3: If min (Yopt ) ¡ 4 or - ¡ 4 then stop (the algorithm has failed). Step 4: If .k − lk ¡ 4 stop iterating (the algorithm has converged); otherwise repeat Step 2 with k = k + 1. If the algorithm has terminated successfully .k → 1, lk → 1 and X → Y −1 and 2 → -−1 . A Lyapunov matrix satisfying (A.3)–(A.4) is given by P = -−1 Y . The required gain M can then be obtained from solving the LMI P(Ap + B2 MC2 ) + (Ap + B2 MC2 )T P + C1T C1 ¡ 0
(A.5)
which (for a given P) is a convex problem.
(A.2)
from the disturbance w is less than a given scalar ,. It is well known that this is equivalent to
D12
References Arzelier, D., Angulo, M., & Bernussou, J. (1997). Sliding surface design by quadratic stabilization and pole placement. Proceedings of the European control conference. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia: SIAM. Chilali, M., & Gahinet, P. (1996). H∞ design with pole placement constraints: An LMI approach. IEEE Transactions on Automatic Control, AC-41, 358–367. Choi, H. (1997). A new method for variable structure control system design: A Linear Matrix Inequality approach. Automatica, 33, 2089–2092. Choi, H. (1999). On the existence of linear sliding surfaces for a class of uncertain dynamic systems with mismatched uncertainties. Automatica, 37, 1707–1715. Edwards, C., & Spurgeon, S. (1996). Robust output tracking using a sliding mode controller/observer scheme. International Journal of Control, 64, 967–983. Edwards, C., & Spurgeon, S. (1997). Sliding mode output tracking with application to a multivariable high temperature furnace problem. International Journal of Robust and Nonlinear Control, 7, 337–351. Edwards, C., & Spurgeon, S. (1998). Sliding mode control: Theory and applications. London: Taylor & Francis. Franklin, G., Powell, J., & Emami-Naeini, A. (2002). Feedback control of dynamic systems (4th ed.). Englewood Cli0s, NJ: Prentice-Hall International Edition. Gahinet, P., Nemirovski, A., Laub, A., & Chilali, M. (1995). LMI control toolbox, user guide. Natick, MA: MathWorks, Inc. Herrmann, G., Spurgeon, S., & Edwards, C. (2001). A robust sliding mode output tracking control for a class of relative degree zero and non-minimum phase plants: A chemical process application. International Journal of Control, 72, 1194–1209. Hung, J., Gao, W., & Hung, J. (1993). Variable structure control: A survey. IEEE Transactions on Industrial Electronics, 40, 2–22. Iwasaki, T., & Skelton, R. (1994). Linear quadratic optimal control with static output feedback. Systems and Control Letters, 23, 421–430.
C. Edwards / Automatica 40 (2004) 1761 – 1769 Ryan, E., & Corless, M. (1984). Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous control. IMA Journal of Mathematical Control and Information, 1, 223–242. Utkin, V. (1992). Sliding modes in control optimization. Berlin: Springer. Utkin, V., Guldner, J., & Shi, J. (1999). Sliding mode control in electromechanical systems. London: Taylor & Francis. Utkin, V., & Young, K.-K. (1978). Methods for constructing discontinuity planes in multidimensional variable structure systems. Automation and Remote Control, 39, 1466–1470. Zinober, A. (1994). An introduction to sliding mode variable structure control. In Zinober, A. (Ed.), Variable structure and capital lyapunov control (pp. 1–22). Berlin: Springer.
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Chris Edwards was born in Swansea, South Wales. He graduated from Warwick University in 1987 with a B.Sc. in Mathematics. From 1987–1991 he was employed as a Research OQcer for British Steel Technical in Port Talbot where he was involved with mathematical modelling of rolling and
Automatica 40 (2004) 1761 – 1769
www.elsevier.com/locate/automatica
Brief paper
A practical method for the design of sliding mode controllers using linear matrix inequalities Christopher Edwards∗ Control and Instrumentation Group, Department of Engineering, University of Leicester, LE1 7RH, UK Received 18 September 2003; received in revised form 14 February 2004; accepted 10 May 2004
Abstract This paper considers the problem of selecting a sliding surface for a given system in an optimal way such that the performance of the reduced order system is balanced against the control costs required to maintain sliding. This approach is quite di0erent from many of the existing schemes in the literature in which no direct account is taken of the control e0ort. Two new numerical schemes are presented both involving convex optimization problems. Three simple examples are given to demonstrate the theory. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Sliding modes; Quadratic cost functions; LMIs
1. Introduction Sliding mode control is an established method of controlling uncertain dynamical systems (Utkin, 1992; Edwards & Spurgeon, 1998). Its invariance properties with respect to so-called matched uncertainty has encouraged researchers to apply sliding mode techniques to a wide variety of application areas (Utkin, 1992; Edwards & Spurgeon, 1998; Utkin, Guldner, & Shi, 1999; Hung, Gao, & Hung, 1993). The early theory was developed within a state-space framework and invariably assumed that full state information was available for use in the control law. The design of a (state feedback) sliding mode controller traditionally involves,
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. ∗ Tel.: +44-116-223-1303; fax: +44-116-252-2619. E-mail addresses: [email protected], [email protected] (C. Edwards). 0005-1098/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.05.004
Linear Matrix Inequality (LMI) methods have been explored (Arzelier, Angulo, & Bernussou, 1997; Choi, 1997, 1999; Herrmann, Spurgeon, & Edwards, 2001). However, from the standpoint of designing a control law for real systems, with limits on the available control action, all these methods have shortcomings. Because of the two-stage nature of the design process, at the time of synthesizing the switching function, no account is taken of the control action that is necessary to induce, and more importantly maintain sliding, even for nominal linear systems. For instance, the optimal quadratic method of Utkin and Young (1978) which resembles the traditional linear quadratic control methods, deliberately has zero penalty on the control e0ort and adopts the principle of ‘cheap control’ (Utkin & Young, 1978; Utkin, 1992). This paper seeks to develop a framework to design the switching function whilst quantifying the control action that is necessary to maintain sliding. This requirement will be formulated as a convex optimization problem consisting of coupled Linear Matrix Inequalities. Linear Matrix Inequalities have been used previously for sliding mode control law design (Arzelier et al., 1997; Choi, 1997, 1999; Herrmann et al., 2001). In Arzelier et al. (1997) LMIs have been used to place the poles of the reduced order sliding motion in convex regions of the complex plane. The emphasis in the work of Choi (1997, 1999) has been largely the design of sliding surfaces so that the sliding motion is robust to unmatched uncertainty. In all these methods no
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explicit account is taken of the resulting control signals required to induce and maintain sliding. In the recent design method of Herrmann et al., 2001, linear quadratic methods are employed and a two-stage optimization is performed. However in terms of the hyperplane design, essentially the method of Utkin and Young (1978) is used (although LMI methods are employed to obtain the solution) and hence no penalties are imposed in terms of the control. This paper is organized as follows: the next section describes the problem that will be addressed in this paper; Section 3 considers two LMI-based methods to synthesize the required sliding surface; Section 4 considers three different simple examples to demonstrate the theory;
(SB)−1 s(t) governs the (asymptotic) rate of convergence to the sliding surface when (·) = 0. This follows since substituting the control law (2) into the nominal linear system (1), the switching function satis
(5)
(1)
(assuming (·) = 0) and thus s(t) → 0 asymptotically. In what follows, as in Choi (1997), it will be assumed that = Im where is a negative scalar. Consequently P2 can be any s.p.d matrix. Without loss of generality it will be assumed that P2 = Im . The control law in (2)–(3) is now entirely parameterized by S and . In designing the sliding surface, i.e. selecting the matrix S, it is usually convenient to assume the system is in regular form (Utkin, 1992). Even though for the method which will be proposed in this paper a di0erent coordinate system will be proposed, as a starting point, it will be assumed that (by a change of coordinates) the system matrices in (1) have the form A11 A12 0 A= ; B= ; (6) Im A21 A22
where the state x ∈ R and the control signal u ∈ R . The unknown signal : R+ ×Rn ×Rm → Rm represents matched uncertainty and is assumed to be norm bounded by a known function (t; x; u). Assume the input distribution matrix B is full rank and the pair (A; B) is controllable. The objective is to design a switching surface of the form
where A11 ∈ R(n−m)×(n−m) . The structure of the input distribution matrix above is more stringent than that normally imposed for regular form, but is necessary for the exposition which follows, and can be achieved very easily. 1 In this coordinate system the switching function matrix may be parameterized so that
S = {x : Sx = 0};
S = S2 [M Im ];
2. Problem statement Consider the uncertain linear system x(t) ˙ = Ax(t) + B(u(t) + (t; x; u)); n
m
where S ∈ Rm×n is a full rank matrix which needs to be designed so that the associated reduced order sliding mode, when the system states are con
(2)
m×m
is a stable matrix and P2 s(t) un (t) = − (t; x; u)(SB)−1 if s(t) = 0; P2 s(t)
(3)
where P2 ∈ Rm×m is a symmetric positive de
(4)
The scalar function (·) must be chosen as an appropriate upper bound on the matched uncertainty satisfying (t; x; u) ¿ SB(t; x; u). Remark 2.1. This is a commonly adopted control structure in the literature (Ryan & Corless, 1984; Zinober, 1994; Edwards & Spurgeon, 1998). The linear part of the control law in (2) can be thought of as comprising a term −(SB)−1 SAx(t) which represents the so-called ‘equivalent control’ necessary to maintain a sliding motion on S (Utkin, 1992) in the absence of uncertainty, whilst the term
m×(n−m)
(7) m×m
where M ∈ R and S2 ∈ R is nonsingular. Theoretically the choice of S2 has no e0ect on the reduced order sliding motion. If (x1 ; x2 ) represents a partition of the states associated with the canonical form in (6), then, from (7), during the sliding motion, s = 0 and so x2 = −Mx1 . Substituting for x2 in the
(8)
and so the sliding motion is thus governed by the system matrix (A11 − A12 M ). If there is no uncertainty in (1) and letting (·) → 0 the control law in (2) becomes linear. The objective is to design the switching function matrix M in (7) so that the control e0ort arising from the nominal linear control law u(t) = −(SB)−1 (SA − S)x(t)
(9)
minimizes a given cost function. This approach of ignoring the matched uncertainty and designing the sliding surface based on the nominal linear model is quite consistent with existing switching function design methods (Utkin, 1992; Zinober, 1994). 1
From a numerical viewpoint it is often convenient to use an orthogonal change of coordinates to achieve regular form (Zinober, 1994; Edwards & Spurgeon, 1998). However in order to obtain the canonical form in (6), a non-orthogonal coordinate change would normally be required.
C. Edwards / Automatica 40 (2004) 1761 – 1769
A plausible formulation is therefore to attempt to choose S and so that the closed-loop system resulting from using the linear control law (9) minimizes the cost functional ∞ x()T Qx() + u()T Ru() d; (10) J=
1763
and hence ˜ = (A11 − A12 M ) ∪ {}m : (A˜ + B˜ L)
where Q ∈ Rn×n is a given symmetric semi-de
This partition of the eigenvalues and the repeated appearance of is expected from the structure of the controller in (9) and the choice of =Im . Indeed this partition of the closed-loop poles and the eigenstructure associated with (5) is inherent in the approaches described in Ryan and Corless (1984); Zinober (1994), Edwards and Spurgeon (1998). The canonical form in (12)–(13) will now be shown to be a useful one in which to synthesize the design freedom encapsulated by M . Two di0erent approaches for the selection of M will now be given in the following subsections.
L = [M Im ](In − A):
3.1. A min–max approach
0
(11)
The parameter M thus represents the design freedom in the problem. Remark 2.2. Although this paper assumes that all the states are available for use in the controller, the scheme could still be implemented if only measured outputs were available, by the use of an observer. As shown in (Edwards & Spurgeon, 1996, 1997), if the nominal system triple relating inputs to measured outputs is relative degree one and minimum phase, by using an appropriate sliding mode observer, all the robustness properties of state feedback sliding mode control laws with respect to matched uncertainty can be retained.
as
Observe that the gain matrix L˜ from (14) can be expressed
L˜ = M [In−m 0(n−m)×m ] + [0 Im ]:
(17)
From this view point the closed-loop system matrix can be written as 0 A11 −A12 A˜ + B˜ L˜ = + MC2 ; (18) A21 Im Im − A22 B2
Ap
where C2 = [In−m 0(n−m)×m ]
3. LMI formulations Now consider the change of coordinates x → (In − A)x. Because, by design, does not belong to the spectrum of A, this is a nonsingular transformation. It will of course (generally) destroy the structure in the input distribution matrix traditionally associated with regular form in (6). In ˜ B) ˜ has the form this new coordinate system the new pair (A; A A 11 12 A˜ = (In − A)A(In − A)−1 = A = ; (12) A21 A22 −A12 B˜ = (In − A)B = : (13) Im − A22 The linear state feedback gain in (11) takes the form L˜ = [M Im ](In − A)(In − A)−1 = [M Im ]
(14)
and the switching function from (7) becomes S˜ = S2 [M Im ](In − A)−1 : It can be readily veri
0 Im
and the design problem becomes a static output feedback LQ problem involving the
(16)
(20)
where B1 = (In − A), w(t) is an exogenous disturbance and the control law is given by u˜ = MC2 x. ˜ The optimization problem of minimizing J from (10) can be formulated as an H2 problem. Let Qq ∈ Rq×n with q 6 n be a full row rank matrix such that QqT Qq = Q from (10). Then formally the equivalent H2 problem can be posed as: Minimize with respect to M the H2 norm of the error signal ˜ z = C1 x + D12 u; where
Qq; 1
C1 : = 0 0
(15)
(19)
(21) Qq; 2
R1=2 0m×m
0q×m
D12 : = 0m×m
(22)
R1=2
and [Qq; 1 Qq; 2 ] := Qq (In − A)−1 . The appearance of the R1=2 term in the C1 matrix comes from the
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C. Edwards / Automatica 40 (2004) 1761 – 1769
function matrix S and control gain L can be obtained from Eqs. (7) and (11). 3.2. A block diagonal approximation As an alternative to the min–max scheme (which requires an iterative solution), a di0erent approach will be considered in this subsection. Again the coordinates associated with (12)–(13) will be used. The minimization of the cost function (10) can be cast as an optimization problem in terms of LMIs, namely: Minimize trace((In − A)T X −1 (In − A)) subject to ˜ + X (A + B˜ L) ˜ T XQqT X L˜T R1=2 (A + B˜ L)X Qq X −Iq 0 ¡ 0; 1=2 ˜ 0 −Im R LX (23) X ¿0
(24)
with respect to the decision variables L˜ and X where X ∈ Rn×n is a s.p.d matrix. Inequality (23) is (e0ectively) inequality 7.40 in Boyd, Ghaoui, Feron, and Balakrishnan (1994). The left and right multiplication of X −1 by (In −A) in the trace operation is required because of the change in coordinates from the special regular form, where the cost function in (10) is de
(n−m)×(n−m)
and X2 ∈ R
m×m
and de
˜ = [N1 X2 ]; N = LX
(26)
m×(n−m)
n×n
. Let Z ∈ R be a s.p.d. matrix of where N1 ∈ R ˜ from (26) in (23) ‘slack’ variables. Then substituting for LX yields the convex optimization problem: Minimize trace(Z) with respect to the variables X; Z and N subject to ˜ + (BN ˜ )T XQqT N T R1=2 AX + XAT + BN Qq X −Iq 0 ¡ 0; R1=2 N
−Z (In − A)
(In − A) −X
0 T
−Im (27)
¡ 0:
(28)
Standard LMI optimization algorithms such as those described in Gahinet, Nemirovski, Laub and Chilali (1995) can be used to obtain a numerical solution to this problem. Once X and N have been synthesized, the gain matrix L˜ can be recovered from the expression L˜ = NX −1 = [N1 X1−1 Im ]
(29)
and thus the design parameter which parameterizes the switching function M := N1 X1−1 : The switching function matrix S and control gain L can be obtained from Eqs. (7) and (11). Although the block diagonal structure enforced on X in (25) potentially introduces conservatism, this structure together with the structure of A˜ + B˜ L˜ in (16) does allow very neatly the introduction of additional constraints (with appropriate LMI representations) on the dynamics of the sliding motion in (8). Taking V (x1 )=x1T X1−1 x1 as a Lyapunov function for the states x1 it follows that pole-placement within standard convex regions of the complex plane such as circles, strips and cones can be incorporated as additional LMI constraints (Chilali and Gahinet, 1996; Gahinet et al., 1995) in terms of the variables X1 and N1 . The LMI representation of these regions will be aQne terms of X1 and N1 and so can be combined with (27)–(28) whilst still retaining a convex optimization problem. In addition, to guard against unmodelled un-matched uncertainties, an inequality of the form A11 X1 + X1 AT11 − A12 N1 − N1 AT12 + DDT + X1 E T EX1 ¡ 0
(30)
can be included where D and E are known matrices of appropriate dimension. This guarantees stability of the reduced order sliding motion in (8) against the e0ects of lumped parametric uncertainty of the form DREx1 (t) for all R ¡ 1. Remark 3.1. • One aspect of both approaches is that the canonical form —so called ‘regular form’—is not used as the basis for the design procedure. • In Herrmann et al. (2001) a two-stage optimization procedure was proposed based on H2 methods. In (Herrmann et al., 2001) the hyperplane (parameterized in terms of M ) was
C. Edwards / Automatica 40 (2004) 1761 – 1769
downside is that the scaled min–max algorithm is more computationally intensive because of its iterative nature 2 and has no convergence guarantees. A bene
(32)
and the state-feedback component of the controller L = [ − 9:8818 − 10:9882]:
(33)
The associated sliding motion is governed by a pole at −0:9882. The LMI computed minimal value for the trace((I2 − A)T X −1 (I2 − A)) with X block diagonal is 109.6819, with √ corresponding upper-bound on the associated H2 norm 109:6819 = 10:4729. With L as given in 2
The diagonalization approach requires the solution of 4n + q + m inequalities in n2 + n decision variables; whilst the min–max algorithm involves 4n + m + q + 5 inequalities in n2 + n + 2 decision variables.
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(33), the true value of the H2 cost can be computed from solving the Lyapunov equation Plq (A + BL) + (A + BL)T Plq = −QqT Qq − LT RL (34) trace(Plq ) = 8:4259. This gives an indifrom which cation of the conservatism introduced by enforcing a block-diagonal structure on X as in (25). For comparison a standard LQR calculation with these choices of Qq and R gives closed-loop poles at −2:2361 ± 2:2361i and a H2 cost of 7:0138. As argued earlier however this is not a feasible solution to the sliding mode controller problem. Using the scaled min–max algorithm (see Appendix) to solve the design problem formulated in Section 3.1 with an upper bound on the square of the H2 norm , = 169, and using initial conditions of I2 =100 and - = 1 the following results are obtained: Iteration
l
.
-
1 2 3 4
0.86448037 0.99680425 0.99685960 0.99684657
150.59570656 1.02019790 1.00065630 1.00117895
0.00574229 0.00562059 0.00560472 0.00558872
The fact that both l → 1 and . → 1 indicates the convergence of the algorithm (Iwasaki & Skelton, 1994). The associated Lyapunov matrix (which is the inverse of the one obtained from the min–max algorithm) 0:6619 −0:7123 X= −0:7123 20:3419 and the static output gain in (18) is given by M = −0:9876 which gives a switching function matrix S = [0:9876 1:0000] and state-feedback gain L = [ − 9:8758 − 10:9876]: The associated sliding motion is governed by a pole at −0:9876. The true value of the H2 cost in this case is 8:4274. This is almost identical to that given by the diagonalization method. It should be noted however that the solution obtained from the min–max algorithm depends (in this case) quite signi
C. Edwards / Automatica 40 (2004) 1761 – 1769
this is conservative: the actual cost obtained from solving (34) is 3.4510.) For comparison a standard LQR calculation with these choices of Qq and R gives closed-loop poles at {−10:0000; −2:0412} and an optimal value of 3:4467. The scaled min–max algorithm with , = 45 and initial conditions I2 =100 and - =1, after 10 iterations, gives a static output feedback gain matrix M = −1:3691. The switching function matrix
States
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L = [ − 13:6906 − 11:3691]: The actual H2 norm is 3:9849 and so in this case there is no improvement over the approximation method described in Section 3.2. Remark 4.1. Because in both case (a) and (b) the cost function matrix Q = QqT Qq is not positive de
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time, sec Fig. 1. Evolution of the states for = 0 and 1.
S = [1:3691 1:0000] and state-feedback gain
1 0.5 0 −0.5 −1 −1.5 −2 0
matrices are 0 1:0000 0 A= 0 0 −0:6000
0
88:7574 ; −24:0000
0 B= 0: 1
The states represent shaft position, shaft speed and a scaling of the current in the armature coils, respectively. 3 The objective here is to minimize the H2 cost problem described earlier, but also to ensure that the complex conjugate pair of poles governing the sliding motion have a damping ratio greater than 0:7660. In order to guarantee this, as argued in Section 3.2, the LMIs (27)–(28), can be augmented with constraints involving X1 and N1 . The sliding motion is governed by the system matrix (A11 − A12 M ) and so, as argued in Chilali and Gahinet (1996), Gahinet et al. (1995) if (A11 X1 + X1 AT11 − A12 N1 − N1T AT12 )sin/ (X1 AT11 − N1T AT12 − A11 X1 + A12 N1 )cos/ (A11 X1 − A12 N1 − X1 AT11 + N1T AT12 )cos/ (A11 X1 + X1 AT11 − A12 N1 − N1T AT12 )sin/
¡0
then the eigenvalues of (A11 −A12 M ) lie in the sector centred at the origin making an angle of / with the negative real axis. To guarantee the damping ratio speci
This scaling is necessary to achieve the special case of regular form given in (6).
Switching function
C. Edwards / Automatica 40 (2004) 1761 – 1769 0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Control
Fig. 2. Evolution of the switching functions for = 0 and 1.
1 0 −1 −2 −3 −4 −5 −6
sary to establish the structure of the input distribution matrix in (6) yielding −0:0186 −0:0065 0:0064 −0:0012 0:0026 −0:1354 0:0020 −0:0028 A= ; −0:1311 0:0349 −0:4684 −0:0095 1:0120 0 0 0 0 B= ; 1 0
Time, sec
C= 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time, sec
Fig. 3. Evolution of the signals for = 0 and 1.
been performed with = 0 and 1. Fig. 1 is the response of the system regulating the angular position output from 1 rad back to zero. Very little di0erence is seen between the response of the linear and nonlinear controllers—particularly in terms of the position output. Fig. 2 shows the evolution of the switching functions. The
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−0:7236
−0:2741
0:6707
−0:1085
0:0043
0:0031
−0:2750
−0:1933
0:0030
0:0212
0
−0:1523
1
:
(35)
1 Choosing Qq = 10 C, R = I2 and = −0:5 the diagonalization procedure produced a sliding surface −0:0434 0:2080 1:0000 0:0000 S= −1:5478 1:0559 0:0000 1:0000
with the reduced order sliding motion governed by poles at {−0:0201; −0:1330}. For this controller, the associated H2 cost is 4:1069. For comparison, if the robust pole-placement algorithm in MATLAB’s Control toolbox is used to assign the reduced order poles at {−0:0201; −0:1330}, using the algorithm described in Section 4 in Edwards and Spurgeon (1998), (which could be described as a ‘traditional’ regular form based approach Utkin & Young, 1978)), the sliding surface matrix −0:0749 0:9924 −1:0000 −0:0000 S= : 0:8582 −0:1643 −0:0000 −1:0000 For the associated linear controller the H2 cost is 4:8829 which is signi
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C. Edwards / Automatica 40 (2004) 1761 – 1769
of the reduced order sliding mode; (c) despite some conservatism, very good results have been obtained on the examples considered. The second approach considers a more general problem but requires the use of the so-called scaled min–max algorithm which is much more computationally intensive. Three examples have been used to demonstrate the eQcacy of the method. Appendix A Consider the system x(t) ˙ = Ap x(t) + B1 w(t) + B2 u(t);
(A.1)
where w(t) is an exogenous disturbance and u(t) is the control input which is assumed to have the static output feedback form u = MC2 x, where M is the control gain to be designed. The objective is to select M so that the H2 norm of the error signal z = C1 x + D12 u
T PAcl + ATcl P + Ccl Ccl ¡ 0
(A.3)
and trace B1T PB1 ¡ ,;
(A.4)
where Acl = Ap + B2 MC2 and Ccl = C1 + D12 MC2 . The scaled min–max algorithm proposed by Iwasaki and Skelton (1994) to synthesize M depends on six variables: the scalars 2; -; .; l ∈ R and the s.p.d. matrices X; Y ∈ Rn×n . The iterative scaled min–max algorithm can be summarized as: Step 1: Choose initial values for Yopt ∈ Rn×n and -opt ¿ 0 and set k = 1. Step 2: Solve the following convex optimization problem: (a) Minimize .k with respect to X and 2 such that 1=2 1=2 In 6 Yopt XYopt 6 .k In
1 6 2-opt 6 .k ⊥ Ap X + XATp B2 C1 X
XC1T −2I
B2 D12
⊥T 6−I
to yield Xopt and 2opt . (b) With these values of Xopt and 2opt solve the following convex optimization problem: Maximize lk with respect to Y and - such that 1=2 1=2 lk In 6 Xopt YXopt 6I
lk 6 2opt - 6 1
C2⊥T (YAp + ATp Y + -C1T C1 )C2⊥
-
-
−I
trace B1T YB1 − ,-
-
-
−I
60
60
to yield Yopt and -opt . Step 3: If min (Yopt ) ¡ 4 or - ¡ 4 then stop (the algorithm has failed). Step 4: If .k − lk ¡ 4 stop iterating (the algorithm has converged); otherwise repeat Step 2 with k = k + 1. If the algorithm has terminated successfully .k → 1, lk → 1 and X → Y −1 and 2 → -−1 . A Lyapunov matrix satisfying (A.3)–(A.4) is given by P = -−1 Y . The required gain M can then be obtained from solving the LMI P(Ap + B2 MC2 ) + (Ap + B2 MC2 )T P + C1T C1 ¡ 0
(A.5)
which (for a given P) is a convex problem.
(A.2)
from the disturbance w is less than a given scalar ,. It is well known that this is equivalent to
D12
References Arzelier, D., Angulo, M., & Bernussou, J. (1997). Sliding surface design by quadratic stabilization and pole placement. Proceedings of the European control conference. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia: SIAM. Chilali, M., & Gahinet, P. (1996). H∞ design with pole placement constraints: An LMI approach. IEEE Transactions on Automatic Control, AC-41, 358–367. Choi, H. (1997). A new method for variable structure control system design: A Linear Matrix Inequality approach. Automatica, 33, 2089–2092. Choi, H. (1999). On the existence of linear sliding surfaces for a class of uncertain dynamic systems with mismatched uncertainties. Automatica, 37, 1707–1715. Edwards, C., & Spurgeon, S. (1996). Robust output tracking using a sliding mode controller/observer scheme. International Journal of Control, 64, 967–983. Edwards, C., & Spurgeon, S. (1997). Sliding mode output tracking with application to a multivariable high temperature furnace problem. International Journal of Robust and Nonlinear Control, 7, 337–351. Edwards, C., & Spurgeon, S. (1998). Sliding mode control: Theory and applications. London: Taylor & Francis. Franklin, G., Powell, J., & Emami-Naeini, A. (2002). Feedback control of dynamic systems (4th ed.). Englewood Cli0s, NJ: Prentice-Hall International Edition. Gahinet, P., Nemirovski, A., Laub, A., & Chilali, M. (1995). LMI control toolbox, user guide. Natick, MA: MathWorks, Inc. Herrmann, G., Spurgeon, S., & Edwards, C. (2001). A robust sliding mode output tracking control for a class of relative degree zero and non-minimum phase plants: A chemical process application. International Journal of Control, 72, 1194–1209. Hung, J., Gao, W., & Hung, J. (1993). Variable structure control: A survey. IEEE Transactions on Industrial Electronics, 40, 2–22. Iwasaki, T., & Skelton, R. (1994). Linear quadratic optimal control with static output feedback. Systems and Control Letters, 23, 421–430.
C. Edwards / Automatica 40 (2004) 1761 – 1769 Ryan, E., & Corless, M. (1984). Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systems via continuous and discontinuous control. IMA Journal of Mathematical Control and Information, 1, 223–242. Utkin, V. (1992). Sliding modes in control optimization. Berlin: Springer. Utkin, V., Guldner, J., & Shi, J. (1999). Sliding mode control in electromechanical systems. London: Taylor & Francis. Utkin, V., & Young, K.-K. (1978). Methods for constructing discontinuity planes in multidimensional variable structure systems. Automation and Remote Control, 39, 1466–1470. Zinober, A. (1994). An introduction to sliding mode variable structure control. In Zinober, A. (Ed.), Variable structure and capital lyapunov control (pp. 1–22). Berlin: Springer.
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Chris Edwards was born in Swansea, South Wales. He graduated from Warwick University in 1987 with a B.Sc. in Mathematics. From 1987–1991 he was employed as a Research OQcer for British Steel Technical in Port Talbot where he was involved with mathematical modelling of rolling and