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Sliding mode control : a tutorial

Sarah Spurgeon1

Abstract— The fundamental nature of sliding mode control where y denotes the angular position and u denotes the is described. Emphasis is placed upon presenting a constructive control, or torque, applied at the suspension point. The scalar theoretical framework to facilitate practical design. The devel- a is positive and when a = 0 the dynamics (1) collapse to opments are illustrated with numerical examples throughout. 1 1 the case of a nominal double integrator. An alternative inter- pretation of equation (1) is that the case a1 = 0 corresponds I.INTRODUCTION to a nominal system and the term −a1 sin(y) corresponds to bounded uncertainty within the nominal dynamics. Define Sliding mode control evolved from pioneering work in a switching function, s which represents idealised dynamics the 1960’s in the former Soviet Union [1], [2], [3], [4]. corresponding to a first order system with a pole at −1 It is a particular type of Variable Structure System (VSS) which is characterised by a number of feedback control s = y˙+ y (2) laws and a decision rule. The decision rule, termed the In the sliding mode, when s = 0, the dynamics of the switching function, has as its input some measure of the system are determined by the dynamicsy ˙ = −y, a free system current system behaviour and produces as an output the where the initial condition is determined by (y(t ),y˙(t )), particular feedback controller which should be used at that s s where t is the time at which the sliding mode condition, instant in time. In sliding mode control, Variable Structure s s = 0 is reached. Defining the sliding mode dynamics by Control Systems (VSCS) are designed to drive and then selecting an appropriate sliding surface is termed as solving constrain the system state to lie within a neighbourhood of the existence problem. A control to ensure the desired sliding the switching function. One advantage is that the dynamic be- mode dynamics are attained and maintained is sought by haviour of the system may be directly tailored by the choice means of solving the reachability problem. A fundamental of switching function - essentially the switching function is requirement is that the sliding mode dynamics must be a measure of desired performance. Additionally, the closed- attractive to the system state and there are many reachability loop response becomes totally insensitive to a particular class conditions defined in the literature [3], [4], [12]. Using the of system uncertainty. This class of uncertainty is called so called η−reachbility condition: matched uncertainty and is categorised by uncertainty that is implicit in the input channels. Large classes of problems of ss˙ < −η|s| (3) practical significance naturally contain matched uncertainty, for example, mechanical systems [5], [6], and this has fuelled it is straightforward to verify that the control the popularity of the domain. u = −y˙− ρsgn(s) (4) A disadvantage of the method has been the necessity to implement a discontinuous control signal which, in theoreti- for ρ > a1 + η where η is a small positive design scalar cal terms, must switch with infinite frequency to provide total ensures the reachability condition is satisfied. Figure 1 shows rejection of uncertainty. Control implementation via approx- the response of the system (1) with the control (4) in the imate, smooth strategies is widely reported [7], but in such nominal case of the double integrator, when a1 = 0 and cases total invariance is routinely lost. There are some impor- in the case of the pendulum, when a1 = 1. The transient tant application domains where a switched control strategy onto the desired sliding mode dynamic is different in each is usual and desirable, for example, in power electronics, and many important applications and implementations have

0.2 been developed [8],[9],[10]. More recent contributions have double integrator 0.1 normalised pendulum

extended the sliding mode control paradigm and introduced 0 the concept of higher order sliding mode control where one −0.1 −0.2

motivation is to seek a smooth control that will naturally and −0.3 velocity accurately encompass the benefits of the traditional approach −0.4 −0.5

to sliding mode control [11]. −0.6 A simple example is the scaled pendulum −0.7 −0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 position y¨ = −a1 sin(y) + u (1) Fig. 1: Phase plane portrait showing the response of the 1 Sarah Spurgeon is with the School of Engineering and Digital Arts, double integrator (a = 0) and the scaled pendulum system University of Kent, Canterbury, UK [email protected] 1 (a1 = 1) with initial conditions y(0) = 1,y˙(0) = 0.1 case, but once the sliding mode is reached both systems first order dynamics when in the sliding mode. It is thus exhibit the dynamics of the free first order system with a intuitively obvious that the effective control action experi- pole at −1. Note that the trajectories appear smooth as the enced by what are two different plants must be different. The discontinuous sgn(s) function in (4) is approximated by the so-called equivalent control represents this effective control s δ smooth approximation |s|+δ where > 0 is small and in the action which is necessary to maintain the ideal sliding motion simulations was taken as δ = 0.01. on S . The equivalent control action is not the control action This tutorial paper seeks to introduce sliding mode control. applied to the plant but can be thought of as representing, Particular emphasis is placed on describing constructive on average, the effect of the applied discontinuous control. frameworks to facilitate sliding mode control design. The To explore the concept of the equivalent control more paper is structured as follows. Section II formulates the formally, consider equation (5) and suppose at time ts the classical sliding mode control paradigm in a state space systems states lie on the surface S defined in (8). It is framework and introduces some of the defining characteris- assumed an ideal sliding motion takes place so that FCx(t) = tics of the approach. A framework for synthesis of classical 0 ands ˙(t) = FCx˙(t) = 0 for all t ≥ ts. Substituting forx ˙(t) sliding mode controllers is described in Section III and from (5) gives Section IV presents a tutorial design example. Section V introduces the concepts of higher order sliding mode control. Sx˙(t) = FCAx(t) + FCBu(t) + FC f (t,x,u) = 0 (11) ≥ II.CLASSICALSLIDINGMODECONTROL for all t ts. Suppose the matrix FC is such that the square matrix FCB is nonsingular. This does not present problems Consider the following uncertain dynamical system since by assumption B and C are full rank and F is a design x˙(t) = Ax(t) + Bu(t) + f (t,x,u) parameter that can be selected. The corresponding equivalent y(t) = Cx(t) (5) control associated with (5) which will be denoted as ueq to demonstrate that it is not the applied control signal, u, is where x ∈ IRn, u ∈ IRm and y ∈ IRp with m ≤ p ≤ n rep- defined to be the solution to equation (11): resent the usual state, input and output. The exposition is − −1 − ξ deliberately formulated as an output feedback problem in ueq(t) = (FCB) FCAx(t) (t,x,u) (12) order to describe the constraints imposed by the availability from (6). The necessity for FCB to be nonsingular ensures of limited state information but the analysis collapses to state the solution to (11), and therefore the equivalent control, is feedback when C is chosen as the identity matrix. Assume unique. The ideal sliding motion is then given by substituting that the nominal linear system (A,B,C) is known and that the the expression for the equivalent control into equation (5): input and output matrices B and C are both of full rank. The ( ) −1 system nonlinearities and model uncertainties are represented x˙(t) = In − B(FCB) FC Ax(t) (13) n m n by the unknown function f : IR+ ×IR ×IR → IR , which is ≥ assumed to satisfy the matching condition whereby for all t ts and FCx(ts) = 0. The corresponding motion is independent of the control action and it is clear that the effect f (t,x,u) = Bξ(t,x,u) (6) of the matched uncertainty present in the system has been n m m nullified. The concept of the equivalent control enables the The bounded function ξ : IR+ ×IR ×IR → IR satisfies inherent robustness of the sliding mode control approach to ∥ξ(t,x,u)∥ < k1∥u∥ + α(t,y) (7) be understood and it is clear from (12) why the total insen- p sitivity to matched uncertainty holds when sliding motion for some known function α : IR+ ×IR → IR+ and positive constant k < 1. The intention is to develop a control law is exhibited. Reconsider the perturbed double integrator in 1 a = which induces an ideal sliding motion on the surface equation (1), this time it will be assumed that 1 0 and that the system is subject to the persisting external perturbation S = {x ∈ IRn : FCx = 0} (8) −0.1sin(t). Figure 2 shows a plot of 0.1sin(t) in relation to the smooth control signal applied to the plant. It is seen that for some selected matrix F ∈ IRm×p. A control law compris- the applied (smooth) control signal replicates very closely ing linear and discontinuous feedback is sought the applied perturbation, even though the control signal is u(t) = −Gy(t) − νy (9) not constructed with a priori knowledge of the perturbation. This property has resulted in great interest in the use of where G is a fixed gain matrix and the discontinuous vector sliding mode approaches for condition monitoring and fault is given by { detection [13]. A key feature of the sliding mode control ρ Fy(t) ̸ approach is the ability to specify desired plant dynamics by ν (t,y) ∥Fy(t)∥ if Fy = 0 y = (10) choice of the switching function. Whilst sliding s = FCx = 0 0 otherwise for all t > ts and it follows that exactly m of the states can be where ρ(t,y) is some positive scalar function. expressed in terms of the remaining n − m. It can be shown The motivating example presented in Section I clearly that the matrix (13) defining the equivalent system dynamics demonstrates that two systems with different dynamics, the has at most n − m nonzero eigenvalues and these are the double integrator and the scaled pendulum, exhibit the same poles of the reduced order dynamics in the sliding mode. 0.25 is nonsingular and the triple (A,B,C) is in the form −Perturbation applied Control action 0.2 [ ] [ ] A11 A12 0 0.15 A = B = A21 A22 B2 0.1 [ ]

0.05 C = 0 T (17)

0 ∈ (n−m)×(n−m) −0.05 where A11 IR and the remaining sub-blocks in

−0.1 the system matrix are partitioned accordingly. Let 0 2 4 6 8 10 time p−m m [ ↔ ↔ ] Fig. 2: The relationship between the smooth control signal F1 F2 = FT applied and the external perturbation once the sliding mode is reached where T is the matrix from equation (15). As a result [ ] FC = F1C1 F2 (18) An alternative interpretation is that the poles of the sliding motion are the invariant zeros of the triple A,B,FC. where ∆ [ ] III.CANONICALFORMFORDESIGN C1 = 0(p−m)×(n−p) I(p−m) (19) This section will consider synthesis of a sliding mode Therefore FCB = F2B2 and the square matrix F2 is nonsingu- control for the system in (5). It is assumed that p ≥ m lar. By assumption the uncertainty is matched and therefore and rank(CB) = m where the rank restriction is required for the sliding motion is independent of the uncertainty. In existence of a unique equivalent control. The first problem addition, because the canonical form in (17) can be viewed which must be considered is how to choose F so that the as a special case of the regular form normally used in associated sliding motion is stable. A control law will then sliding mode controller design, the reduced-order sliding be defined to guarantee the existence of a sliding motion. motion is governed by a free motion with system matrix s ∆ − −1 A11 =A11 A12F2 F1C1 which must therefore be stable. If ∈ m×(p−m) −1 A. Switching Function Design K IR is defined as K = F2 F1 then In view of the fact that the outputs will be considered, it s − A11 = A11 A12KC1 (20) is first convenient to introduce a coordinate transformation to make the last p states of the system the outputs. Define and the problem of hyperplane design is equivalent to a [ ] static output feedback problem for the triple (A11,A12,C1). T Nc Appealing to established results from the wider control Tc = (14) C theory, it is necessary that the pair (A11,A12) is controllable and (A ,C ) is observable. It is straightforward to verify that n×(n−p) 11 1 where Nc ∈ IR and its columns span the null space of (A11,A12) is controllable if the nominal matrix pair (A,B) C. The coordinate transformation x 7→ Tcx is nonsingular by is controllable. To investigate the observability of (A11,C1) construction and, as a result, in the new coordinate system partition the submatrix A11 so that [ ] [ ] C = 0 Ip A1111 A1112 A11 = (21) A1121 A1122 From this starting point a special case of the so-called (n−p)×(n−p) regular form defined for the state feedback case [14] will where A1111 ∈ IR and suppose the matrix pair be established. Suppose (A1111,A1121) is observable. It follows that [ ]   [ ] Bc1 ↕n−p zI − A1111 A1112 B = zI − A11   Bc2 ↕p rank = rank A1121 zI − A1122 C1 0 Ip−m Then CB = B and so by assumption rank(B ) = m. Hence [ ] c2 c2 zI − A the left pseudo-inverse B† = (BT B )−1BT is well defined = rank 1111 + (p − m) c2 c2 c2 c2 Ao and there exists an orthogonal matrix T ∈ IRp×p such that 1121 [ ] for all z ∈ C and hence from the Popov-Belevitch-Hautus T 0 T Bc2 = (15) (PBH) rank test and using the fact that (A1111,A1121) is B2 observable, it follows that [ ] ∈ m×m where B2 IR is nonsingular. Consequently, the coordi- zI − A11 7→ rank = n − m nate transformation x Tbx where C1 [ ] − † ∈ In−p Bc1Bc2 for all z C and hence (A11,C1) is observable. If the Tb = T (16) 0 T pair (A1111,A1121) is not observable then there exists a ∈ (n−p)×(n−p) o Tobs IR which puts the pair into the following The spectrum of A11 represents the invariant zeros of the observability canonical form: nominal system (A,B,C). [ ] o o − A A From discussion of the canonical form it follows that there T A T 1 = 11 12 S obs 1111 obs 0 Ao exists a matrix F defining a surface which provides a [ 22] stable sliding motion with a unique equivalent control if and A T −1 = 0 Ao 1121 obs 21 only if o ∈ r×r o ∈ (p−m)×(n−p−r) o o where A11 IR , A21 IR , the pair (A22,A21) • the invariant zeros of (A,B,C) lie in C− ≥ is completely observable and r 0 represents the number • the triple (A˜11,B˜1,C˜1) is stabilisable by output feedback. (A ,A ) of unobservable states of 1111 1121 . The transformation The first condition ensures that any invariant zeros, which T can be embedded in a new state transformation matrix obs [ ] will appear within the poles of the reduced order sliding Tobs 0 motion, are stable. The second condition ensures the reduced Ta = (22) 0 Ip order sliding mode dynamics are rendered stable by the which, when used in conjunction with Tc and Tb from choice of sliding surface. Although these conditions may equations (14) and (16), generates the required canonical appear restrictive, techniques such as sliding mode differ- form. entiators [15], or other soft sensors, and the availability of 1) Canonical Form for Sliding Mode Control Design: inexpensive hardware sensors can be helpful in ensuring Let (A,B,C) be a linear system with p > m and rank(CB) = sufficient information is available to tailor the reduced order m. Then a change of coordinates exists so that the system dynamics in the sliding mode. triple with respect to the new coordinates has the following structure: B. Reachability of the Sliding Mode • The system matrix can be written as Having established a desired sliding mode dynamics by [ ] ′ ∈ m ×(p−m) ˜ − ˜ ˜ A11 A12 selecting K1 IR such that A11 B1K1C1 is stable A f = (23) A21 A22 and providing any invariant zeros are stable, it follows that ∈ (n−m)×(n−m) λ − λ o ∪ λ ˜ − ˜ ˜ where A11 IR and the sub-block A11 when (A11 A12KC1) = (A11) (A11 B1K1C1) partitioned has the structure   − o o and so the matrix A11 A12KC1 determining the dynamics A11 A12 m  o A12  in the sliding mode is stable. Choose A11 = 0 A22 (24) [ ] o m T 0 A21 A22 F = F2 KIm T × − − × − − Ao ∈ r r Ao ∈ (n p r) (n p r) Ao ∈ × where 11 IR , 22 IR and 21 where nonsingular F ∈ IRm m must be selected. Introduce a (p−m)×(n−p−r) ≥ ( o , o ) 2 IR for some r 0 and the pair A22 A21 nonsingular state transformation x 7→ Tx¯ where is completely observable. [ ] • The input distribution matrix has the form I − 0 [ ] T¯ = (n m) (29) 0 KC1 Im B f = (25) B2 and C1 is defined in (19). In this new coordinate system the m×m ¯ ¯ ¯ where B2 ∈ IR and is nonsingular. system triple (A,B,FC) has the property that • [ ] [ ] The output distribution matrix has the form ¯ ¯ [ ] ¯ A11 A12 ¯ 0 C = 0 T (26) A = ¯ ¯ B = f [ A21 A]22 B2 p×p where T ∈ IR and is orthogonal. FC¯ = 0 F2 (30) In the case where r > 0, it is necessary to construct a where A¯11 = A11 − A12KC1 and is therefore stable. An alter- new system (A˜11,B˜1,C˜1) which is both controllable and observable with the property that native description is that by an appropriate choice of F a new square system (A¯,B¯,FC¯) has been synthesised which is λ s λ o ∪ λ ˜ − ˜ ˜ (A11) = (A11) (A11 B1KC1) minimum phase and relative degree 1. m Let P be a symmetric positive definite matrix partitioned Partition the matrices A12 and A12 as follows [ ] [ ] conformably with the matrices in (30) so that A Am A = 121 and Am = 121 (27) [ ] 12 A 12 Am P 0 122 122 P = 1 (31) ∈ (n−m−r)×m m ∈ (n−p−r)×(p−m) 0 P2 where A122 IR and A122 IR . Form a new sub-system (A˜ ,A ,C˜ ) where 11[ 122 1 ] where the symmetric positive definite sub-block P2 is a ∆ Ao Am design matrix and the symmetric positive definite sub-block A˜ = 22 122 11 Ao Am P1 satisfies the Lyapunov equation [ 21 22 ] ˜ ∆ ¯ ¯T − C1 = 0(p−m)×(n−p−r) I(p−m) (28) P1A11 + A11P1 = Q1 (32) ∆ T Root Locus for some symmetric positive definite matrix Q1. If F =B2 P2 6 then the matrix P satisfies the structural constraint PB¯ = 4 C¯TFT For notational convenience let

) 2 ∆ −1 Q = P A¯ + A¯T P 2 1 12 21 2 (33) 0

∆ T Imaginary Axis (seconds ¯ ¯ −2 Q3 = P2A22 + A22P2 (34) and define −4 ( ) ∆ −6 − − − −10 −8 −6 −4 −2 0 2 1 1 T T 1 1 −1 γ λ Real Axis (seconds ) 0 = 2 max (F ) (Q3 + Q2 Q1 Q2)F (35) ∆ T Fig. 3: Variation in the closed-loop poles with the switching The symmetric matrix L(γ)=PA0 + A P where A0 = A¯ − 0 surface parameter K γBF¯ C¯ is negative definite if and only if γ > γ0. A control law to induce sliding on S is given by equations (9-10) with G = γF and γ > γ0 where γ0 is defined in (35). The uncertain From (23) and (28) system (5) is quadratically stable and an ideal sliding motion   S 0.091 −8.832 −0.739 is induced on .   A11 = 0.592 0.279 −2.292 IV. NUMERICAL EXAMPLE 0 10.803 −0.329 [ ] Consider the nominal linear system representing the longi- AT = −11.401 27.719 3.106 tudinal dynamics of a fixed wing Unmanned Aerial Vehicle 12 [ ] (UAV) developed for medium-range autonomous flight mis- C1 = 0 0 1 (40) sions including search and rescue, weather monitoring, aerial The design requirement is to determine K such that photography and reconnaissance [16]. The states represent A − A KC has desired dynamics. The root locus plot deviations from nominal forward speed u (m/s), vertical 11 12 1 for the sub-system (40) is shown in Figure 3. The loca- velocity w (m/s), pitch rate q (rad/s) and pitch angle θ(rad) tion of the open-loop pole at −0.140 which moves to the and the control signal is the elevator deflection η (rad) [16]:   zero at 0.140 limits the acceptable dynamic performance; −0.218 −0.225 4.990 −9.184   as gain is increased the system becomes unstable. Select-  −0.137 −0.233 10.592 −2.984  ing K = 1 prescribes the poles of the sliding motion at A =   0.009 −0.070 −3.282 −0.566 −0.1149,−1.1043,−26.2944 and thus improves the stability 0 −0.002 0.969 −0.014 [ ] margin of the initial design whilst reducing the speed of the BT = 1.754 2.301 −4.741 −0.063 (36) fastest pole. In the original coordinates [ ] The poles of the open-loop system are located at F = 11.252 − 23.832 (41) −2.778,−0.044 ± 0.1587 j,−0.881. Consider first the situ- ation where the pitch angle, θ only is measured. θ η [ ] Figure 4 show the response of , and s using the control γ γ Cθ = 0 0 0 10 (37) (9-10) with G = F and = 123.0. In this case a single-input single-output system results and there are no degrees of freedom available to design the −3 x 10 15 sliding surface, which in this case is wholly defined by the nominal output equation. The dynamics in the sliding mode are given 10 perturbed 5 by the transmission zeros of the triple (A,B,Cθ ) which can be computed to be −76.331,−0.419,−0.056. theta (rad) 0 −5 Consider now the case where both u and θ are measured: 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 [ ] time (s) 1 0 0 0 4 Cu,θ = (38) nominal 0 0 0 10 2 perturbed In this case the system has no transmission zeros, the number

eta (deg) 0 of measurements exceeds the number of controls and there −2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 is some design freedom available to tailor the sliding mode time (s) dynamics. In the canonical form (23-26) 0.01   nominal 0.091 −8.832 −0.739 −22.594 0 perturbed  0.592 0.279 −2.292 1.157  s −0.01 A =   f  0. 10.803 −0.329 27.555  −0.02 −0.03 −0.194 −1.429 0.879 −3.788 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 [ ] time (s) T − B f = 0 0 0 1.864 [ ] Fig. 4: Response of the perturbation in theta, elevator angle 0 0 0.338 −0.941 C = (39) and switching function for both a nominal and perturbed f 0 0 0.941 0.338 model of the UAV with initial condition x(0) = [0 0 0.5 0]T V.HIGHERORDERSLIDINGMODES 0.4 Thus far the focus has been on enforcing a sliding mode on 0.2 S by applying a discontinuous injection to thes ˙ dynamics. 0 As has been previously mentioned, a key disadvantage is −0.2 the fundamentally discontinuous control signals result. The velocity −0.4 concept of Higher Order Sliding Modes (HOSM) generalise −0.6 the sliding mode control concept so that the discontinuity −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 acts on higher order derivatives of s and the applied control is position smooth. In general, if the control appears on the rth derivative of s, the rth order ideal sliding mode is defined by: Fig. 5: Phase plane portrait of (1) with a1 = 1, y(0) = 1,y˙(0) = 0.1 and the super-twisting controller s = s˙ = s¨ = .... = s(r−1) = 0 (42) In fact if the control is implemented with sample period continuous time, discontinuous control strategies fundamen- T, | s |= O(T r),...| s(r−1) |= O(T r). The total invariance to tally rely upon very high frequency switching to ensure the matched uncertainty exhibited by the traditional sliding mode sliding mode is attained and maintained. The introduction of control holds for higher order sliding modes as well as finite sampling is disruptive. For example, switching of increasing time convergence to the sliding surface. Perhaps the most amplitude can take place about the sliding surface. Reviews commonly implemented higher order sliding mode control is of the discrete time sliding mode control paradigm can be the super-twisting algorithm which is a second order sliding found in [17] [18]. mode control. Consider the sliding variable dynamics REFERENCES s˙ = ϕ(s,t) + γ(s,t)ust (43) [1] U.Itkis, Control systems of variable structure, Wiley, New York, 1976. [2] V.I.Utkin, Variable structure systems with sliding modes, IEEE Trans- with |ϕ| ≤ Φ and 0 ≤ Γm ≤ γ(s,t) ≤ ΓM. The super-twisting actions Automatic Control, 22, 1977, pp. 212-222. controller is defined by [3] V.I.Utkin, Sliding modes in Control Optimisation, Springer-Verlag, Berlin, 1992. ust = u1 + u2 (44) [4] C.Edwards and S.K.Spurgeon, Sliding mode control: Theory and { Applications, Taylor and Francis, 1998. −ust if |ust | > U [5] V. Utkin, J. Guldner, J. 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