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154

A Hybrid Sliding Mode Strategy for Minimum-Phase Systems with Output

Constraints

Hanz Richter

Abstract— A new setpoint control scheme is proposed forsingle-input systems, where a regulated output must be trans-ferred between two setpoints, with the additional requirementthat a set of limited outputs remain within prescribed bounds.The strategy is based on a max-min selector system frequentlyused in the process control and aerospace fields. The regulatorsused for the controlled and outputs are of the sliding mode type,where the sliding variable is defined as the difference betweenan output and its limit. The max and min arrangements enjoymany beneficial properties, including a separation principleallowing the independent design of the sliding regulators,

with overall asymptotic stability guarantee. The paper detailsthe stability proof for the separate min and max selectorsystems and describes the invariance properties leading to limitprotection. Some results applicable to the max-min case are alsoincluded. Design guidelines, as well as a detailed simulationexample, demonstrate practical applicability.

I. INTRODUCTION

This paper considers an aspect of the general problem of

feedback control of systems with constraints, in particular

the problem of output inequality constraints. A new setpoint

control scheme is proposed for single-input systems, where

a “main” or “regulated” output must be transferred between

two setpoints, with the additional requirement that a set of

“auxiliary” or “limited” outputs remain within prescribed

limits. Limit protection under nominal and steady-state con-

ditions is straightforward to achieve when solutions are avail-

able for equilibrium points. Given a fixed controller, one has

to restrict the range of the setpoint of the regulated output, so

that the corresponding steady values of the auxiliary outputs

remain within the desired limits. Limit violation, thus, will

ensue at steady state if the regulated variable is driven to a

setpoint outside the permissible range. There are two other

circumstances that cause limits to be exceeded, namely tran-

sient violations and off-nominality. Indeed, limit preservation

at steady-state does not imply that auxiliary variables will

remain within limits during their transition to steady-state.

The prevention of such transient violations is not easily

included in the controller design process. Off-nominality may

arise from a change in controller tuning to address new

performance requirements for the regulated variable, or from

unintentional factors such as plant parameter variations or

external disturbances.

The requirement that outputs of the form y = Cx+Du re-

main within certain bounds at all times may be reduced to the

problem of control with state variable and input constraints,

topic for which a vast amount of literature can be found. For

Department of Mechanical Engineering, Cleveland State University,Cleveland, OH 44115, USA [email protected]

1

s

x = Ax + Bu

yj = Gjx + Θjumax

min

f1(x, u)

f2(x, u)

fl(x, u)

fl+1(x, u)

fl+2(x, u)

fh(x, u)

x1, u1

x2, u2

xl, ul

xl+2, ul+2

xh, uh

xl+1, ul+1

Fig. 1. A Max-Min Selector Control Scheme with Input Integration

an excellent survey, see [4]. These set invariance-based meth-

ods typically suffer from a tradeoff between conservativeness

(i.e., designed controllers do not fully exploit the available

limits) and computational complexity (i.e., a large number

of decision inequalities must be evaluated in real-time). In

the case of state-constrained systems controlled with sliding

modes, invariant set techniques have been developed in [10].

Similarly, constraint enforcement via model-predictive

control, although effective, is widely-regarded as a

computationally-intensive process, although efforts have

been made to reduce its complexity [2], [3], [11]

When only output limiting is sought, a number of “limit

protection logic” schemes have been successfully used for

decades in the aerospace industry, but without sufficient

supporting analyses characterizing their behavior, including

the essential discrimination between successful and failed

implementations. One popular arrangement for a single out-

put which must be precisely regulated while keeping other

outputs within limits is the multi-regulator scheme with a

min-max selection criterion shown in Fig. 1. One of the

regulators f(x, u) is intended for the main output, while

the others are limit regulators. Due to the requirement that

the main output be precisely regulated, integration is used

at plant input. The regulators, thus, provide control input

rates. The control rate applied to the integrator is selected

as the maximum over the rates produced by the lower-limit

regulators and the minimum over those produced by the

upper-limit regulators. Letting L = {1, 2, ...l} and H ={l + 1, l + 2, ...h}, the max-min selection law is expressed

as

ur = maxk∈H

{

minj∈L

{urj} , urk

}

(1)

where urj are the lower-limit regulator outputs and urk are

the upper-limit regulator outputs. Some studies character-

izing the behavior of related schemes have appeared, for

978-1-4244-5831-8/10/$26.00 ©2010 IEEE

2010 11th International Workshop on Variable Structure Systems

Mexico City, Mexico, June 26 - 28, 2010

155

instance in [1], [5], [7]. More recently, the (non-asymptotic)

stability of this particular scheme under linear regulators has

been analyzed by Johansson [8] using piecewise-quadratic

Lyapunov functions. Even for linear regulators, a complete

characterization of closed-loop behavior does not exist which

includes essential issues such as determining which regulator

will be active at steady-state or how to design the regu-

lators to address performance requirements. In the context

of aircraft engine control, where limit protection is indis-

pensable, few works addressing the max-min arrangement

have appeared [9], [12]. Of particular importance is the

observation that limit regulators may become active even

when the auxiliary outputs are far from their limits, causing

a degradation in the response of the main output due to an

overriding control objective [9].

II. PLANT DESCRIPTION, ASSUMPTIONS AND CONTROL

OBJECTIVES

We consider linear single-input plants with integrated input

given by the state-space description

x = Ax + Bu (2)

u = ur (3)

where x ∈ Rn, and u and ur are scalars. Assume that a set

of outputs is defined as

yi = Gix + Θiu (4)

for i = 1, 2, ...h, with with Gi an 1-by-n vector and Θi an

scalar. We make the following assumptions:

Assumption 1: A is non-singular.

Assumption 2: Define

Aeq,i = A −BGi

Θi

(5)

Then Θi 6= 0 and matrices Aeq,i have eigenvalues with

negative real parts, for i = 1, 2, ...h, that is, yi are minimum-

phase outputs of Eq. 2.

Note that when Assumption 1 fails due to a single zero

eigenvalue a straightforward modification of the results of

this paper extends its applicability by elimination of input

integration. The case Θi = 0 is addressed in another

publication by the author.

III. BEHAVIOR UNDER A FIXED REGULATOR

Define the sliding variables as si = yi − yi, for i =1, 2, ...h, where yi = Gixi + Θiui. The reference variables

xi and ui are selected to be equilibrium pairs, that is, so that

Axi+Bui = 0. Each sliding mode regulator is independently

specified as customarily done, that is, so that the sliding

variable converges to zero in finite time and that a stable

sliding mode ensues. The widely-used control law

uri = −1

Θi

(Gi(Ax + Bu) + ηi sign (si)) (6)

is used as the control rate output of the i-th regulator. Let

i and j be two regulator indices and define the augmented

state as xa , [xT |u]T , let xai = [xTi |ui]

T and define the

augmented state relative to i as xa , xa − xai. Using

this definition, it is straightforward to derive the following

identities pertaining to system behavior under control law 6:

˙xa = Aixa −1

Θi

Biηi sign (si) (7)

sj = Jj xa + ∆j,i (8)

sj|i = Θj

(

Γj,ixa −ηi

Θi

sign (si)

)

(9)

where:

Ai =

[

A B

−Gi

ΘiA −Gi

ΘiB

]

(10)

BTi = [01×n|1] (11)

Jj = [Gj |Θj] (12)

∆j,i = Jj(xai − xaj) (13)

Γj,i =

[

Gj

Θj

−Gi

Θi

]

[

A∣

∣B]

(14)

The notation sj|i should be interpreted as “the derivative of

sj when i is the active regulator”. When i = j we simply

write si. Note that ∆i,i = 0 and Γi,i = 0 for i = 1, 2, ...h.

It is a standard fact of sliding mode theory that for each

i, the spectrum of Ai is formed by the eigenvalues of the

Aeq,i from Eq. 5 and zero. The closed-loop system resulting

from applying input 6 to system 2, 3 is more conveniently

described in terms of the derivatives of the s variables, as

before, and the rate of x. In fact, define Xr , x. The closed-

loop system dynamics are expressed as

Xr = Aeq,iXr − Bηi

Θi

sign (si) (15)

sj|i = Θj

[(

Gj

Θj

−Gi

Θi

)

Xr −ηi

Θi

sign (si)

]

(16)

The rate system is a convenient description, due to Aeq,i

being Hurwitz. It allows to describe asymptotic properties.

However, si cannot be written as a function of Xr in a

manner analogous to Eq.8. For this reason, both descriptions

of the closed-loop dynamics will be used for different

purposes, as convenient.

A. Characterization of the Equilibrium Point

The minimum (min), maximum (max) switching functions

are expressed by Eqs. 17 and 18, respectively.

q = argmini∈L

{uri} (17)

q = argmaxj∈H

{urj} (18)

If q is not uniquely determined by the above equations, it

may be assigned a value according to an arbitrary rule. For

the remainder of this paper, q = min (i, j) is assumed to

be selected when uri = urj . This assumption applies to

the min and max selector arrangements. When a max-min

arrangement is used, it is assumed that the min preselection

is applied to the first port of the max selector, so that the

min input is used in case of equality to the max preselection.

These assumptions will be referred to as default index



156

assumptions. As usual, System (2, 3) is said to have an

equilibrium point at (x, u) if both state derivatives vanish,

that is, Ax + Bu = 0 and ur = 0.

Proposition 1: Under the min switching law, system (2, 3)

has a unique equilibrium point at (xi∗ , ui∗), where i∗ ∈ L is

the index s.t.sign (∆j,i∗ )

Θj≤ 0 ∀j ∈ L. This index is termed

terminal regulator index.

Proof: The requirement that u = 0 implies that urq =0, where q is the index of the active regulator. From Eq. 6, it

is clear that sq = 0 at equilibrium. Now, for q to be the active

regulator, it is necessary that urq ≤ urj for all j ∈ L. At

equilibrium this implies that q is such that − ηq

Θqsign (sq) =

0 ≤ − ηj

Θjsign (∆j,q) for all j ∈ L. Since this inequality

amounts to the selection of the minimum of a set of numbers

with a default selection applicable in the case of equality,

index q = i∗ is uniquely defined and it always exists. Define

now xa relative to i∗. It is clear that ˙xa = 0 at equilibrium.

Using Eqs. 7 and 8 together with the facts that all (xi, ui)are equilibrium pairs and that Aeq,i∗ is nonsingular, we have

xa = 0, implying that (xi∗ , ui∗) is the equilibrium point.

The definition of i∗ in terms of ∆j,i∗ is also clear.

Note that the existence of a time tr ≥ 0 such that q(t) = i∗

for all t ≥ tr has not been yet established. Indeed, the

fact that q remains constant once x = 0 = u is shown

later. The computation of the terminal regulator index is

straightforward.

IV. STABILITY ANALYSIS: MIN SWITCHING

Unlike single-regulator sliding mode control schemes, one

may not use s2q for some fixed q as a Lyapunov function

showing global attractiveness of the set sq = 0. Even

when q = i∗, it is easy to find a simulation counter-

example showing that s2q is non-monotonically decreasing

towards zero. Moreover, the multiple Lyapunov approaches

of Branicky [6], and even the much less restrictive approach

of [13], are difficult to apply. In the first case, one must prove

that s2j is monotonically decreasing in intervals during which

q(t) = j and that the sequence s2(tj) is decreasing, where

tj is the sequence of times at which j is switched on. In the

second case, monotonicity is no longer required during the

active intervals, and the s2(tj) may be increasing, but must

be bounded by a function satisfying certain requirements.

In this paper, a proof of global asymptotic convergence to

the equilibrium point xai is developed which relies only on

Assumptions 1 and 2. The proof is based on attractive-

ness properties of each individual sliding set, together with

considerations about the geometry of the regions of Rn+1

in which each regulator is active under any of the three

switching laws.

Given a fixed i ∈ L, define the following sets

R+i = {xa|uri ≤ urj, si > 0, ∀j ∈ L} (19)

R−i = {xa|uri ≤ urj, si < 0, ∀j ∈ L} (20)

R0i = {xa|uri ≤ urj, si = 0, ∀j ∈ L} (21)

Rj = Rn+1 \ (R+

i ∪R−i ∪R0

i ) (22)

where \ denotes set difference.

Proposition 2: The collection{

R+i∗ ,R

−i∗ ,R0

i∗ ,Rj

}

is a

partition of Rn+1 and 0 ∈ R0

i∗ .

Proof: Clearly, Rj ∪R+i∗ ∪R−

i∗ ∪R0i∗ = R

n+1. Also,

si∗ = 0 defines a hyperplane dividing Rn+1 into three

disjoint regions corresponding to si∗ > 0, si∗ = 0 and

si∗ < 0, therefore all four sets are pairwise disjoint, showing

the partition property. To show that 0 ∈ R0i∗ , note that

si∗ = 0 whenever xa = 0. Since i∗ is the active regulator,

uri∗ ≤ urj∀j ∈ L and thus 0 ∈ R0i∗ .

Lemma 1: The following statements hold:

1) The set R0i∗ is invariant, that is, if ∃tr ≥ 0 s.t. xa(tr) ∈

R0i∗ , then xa(t) ∈ R0

i∗ for t ≥ tr. In addition, xa(t) →0 as t → ∞.

2) If xa(t1) ∈ R+i∗ ∪R−

i∗ for some t1 ≥ 0, then ∃t2 ≥ t1s.t. xa(t2) ∈ R0

i∗ .

3) If xa(t1) ∈ Rj for some t1 ≥ 0, then ∃t2 ≥ t1 s.t.

xa(t2) ∈ R+i∗ ∪R−

i∗ ∪R0i∗ .

Proof: Proof of 1: Suppose xa(tr) ∈ R0i∗ for some

tr ≥ 0. Then si∗(tr) = Ji∗ xa(tr) = 0 and si∗(tr) =ηi∗ sign (si∗(tr)) = 0. Also, uri∗(tr) ≤ urj(tr). We need

to verify that uri∗(t) ≤ urj(t) and si∗(t) = 0 for t ≥ tr.

Two possibilities exist: either there is no mode change, i.e.,

q(t) = i∗ for t ≥ tr, or a mode change exists at some

time t1 > tr. Suppose, first, that q(t) = i∗ (and therefore

si∗(t) = 0) for tr ≤ t < t1 and that q(t1) = j for some

j 6= i∗. Since sk are continuous functions of time for all k,

we must have that si∗(t1) = 0. The change from mode i∗ to

mode j at t1 requires that uri∗(t) ≥ urj(t) for t → t+1 and

uri∗(t) ≤ urj(t) for t → t−1 . With a slight abuse of notation,

this means

Γj,i∗ xa(t−1 ) ≤ −ηj

Θj

sign (sj(t−1 )) (23)

Γj,i∗ xa(t+1 ) ≥ −ηj

Θj

sign (sj(t+1 )) (24)

Since the left-hand side of inequalities 23, 24 is also contin-

uous at t1 and ηj > 0, it is necessary that

−1

Θj

sign (sj(t−1 )) ≥ −

1

Θj

sign (sj(t+1 )) (25)

Although sj is continuous, it is the argument of a discontinu-

ous function. It can be directly verified that strict inequality

in 25 can only be satisfied if sj(t1) = 0, which implies

si∗|j(t1) = 0 from Eqs. 8 and 9. If equality is considered,

we also have, using Γi∗,j xa = −Γj,i∗ xa in Eq. 9, that

si∗|j(t1) = 0. Thus, s∗i (t) = 0 for tr ≤ t ≤ t1 and it can be

directly verified that the condition uri∗(t1) ≤ urj(t1) holds,

contradicting the assumption that q(t1) = j 6= i. Therefore

q(t) = i∗ and si∗(t) = 0 for t ≥ tr, verifying invariance.

Asymptotic convergence of Xr to zero is immediate from

Eq. 15. Since si∗ = Ji∗ xa = 0 for t ≥ tr and Xr =[A|B]xa, it is evident that xa(t) → 0 as t → ∞.

Proof of 2: Suppose xa(t1) ∈ R+i∗ ∪ R−

i∗ for some t1 ≥0. Since R+

i∗ ∩ R−i∗ = ∅, suppose first that xa(t1) ∈ R+

i∗ .

This implies q(t1) = i∗. If there is no mode change, that is,

q(t) = i∗ for t ≥ t1, we have si∗(t) = −ηi∗ sign (si∗(t))so ∃t2 ≥ t1 s.t. si∗(t2) = 0, that is, xa(t2) ∈ R0

i∗ . Suppose



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now, by contradiction, that q(t′) = j 6= i∗ for some t′ ≥ t1.

Following an argument analogous to the proof of the first

statement of this Lemma, it can be deduced that sj(t′) = 0

and that si∗|j(t′+) = 0. Noting that

uri∗ − urj = Γj,i∗ xa +ηj

Θj

−ηi∗

Θi∗

Eq. 9 shows that uri∗(t′+) = urj(t

′+), contradicting the

existence of a mode change from i∗ to j at t = t′. Therefore

the only possibility is that xa(t2) ∈ R0i∗ for some finite

t2 ≥ t1.

Proof of 3: Proceeding by induction, we first show that

the statement is true for l = 2. Suppose xa(t1) ∈ Rj

for some t1 ≥ 0. Then q(t1) = j 6= i∗. Suppose, by

contradiction, that q(t) = j for all t ≥ t1. Then the system

would behave as if it were controlled by single sliding-

mode regulator, reaching the set sj = 0 in finite time and

entering a sliding mode defined by Eq. 15. It can be directly

verified that xaj would meet the definition of an equilibrium

point. If xaj = xai∗ , then ∆j,i∗ = 0 and s∗i = 0, so

j and i∗ would both satisfy the min selection conditions,

since uri∗ = urj . This contradicts the uniqueness of the

terminal regulator index. If xaj 6= xai∗ , the uniqueness of the

equilibrium point is contradicted. Therefore, q must switch to

i∗ at some finite t2 ≥ t1, implying xa(t2) ∈ R+i∗∪R

−i∗∪R

0i∗ .

Now suppose that the statement holds for l − 1 regulators.

We wish to show that it must hold for l regulators. Define

Ll−1 = {1, 2, ...l − 1} . Let k∗ denote the terminal index

in Ll−1 and i∗ denote the terminal index in Ll. Suppose

xa(t1) ∈ Rj for some t1 ≥ 0. Then q(t1) = j 6= i∗ for

some j ∈ Ll. Suppose, again by contradiction, that q(t) = j

for all t ≥ t1. Then the system would behave as if only l−1regulators existed. By the inductive hypothesis, q(t2) = k∗

for some finite t2 ≥ t1, implying xa(t2) ∈ R+k∗ ∪R

−k∗ ∪R0

k∗ .

We have thus shown that, given any set of indices q must

eventually switch to the terminal index in that set, concluding

the proof.

Theorem 1: Under Assumptions 1 and 2, all trajectories

of System (2, 3) under control input 6 and the min switching

law converge asymptotically to the unique equilibrium point

xai∗ .

Proof: Suppose first that xa(0) ∈ R0i∗ . Then the first

statement of Lemma 1 proves the theorem. Now suppose

that xa(0) ∈ R+i∗ ∪ R−

i∗ . The second and first statements

of Lemma 1 are applied to prove asymptotic stability. The

only remaining possibility is, by the partition property of

Proposition 2, that xa(0) ∈ Rj . The third statement of

Lemma 1 establishes that q = i∗ in finite time. If si∗ = 0at this time, the first statement proves asymptotic stability.

If not, the second statement followed by the first proves the

desired result.

A. Stability Under Max Switching

Noting that max{uri} = −min{−uri}, it is evident that

the stability proof holds under max switching, since the

validity of the statements of Lemma 1 are independent of

the direction of the inequalities used in the proofs. The

terminal regulator index, however, is computed differently,

as established next.

Proposition 3: Under the max switching law, sys-

tem (2, 3) has a unique equilibrium point at (xi∗ , ui∗), where

i∗ ∈ H is the index s.t. sign (∆j,i∗) ≥ 0 ∀j ∈ H . This

index is termed terminal regulator index.

B. Stability Under Max-Min Switching

The max-min case requires additional analysis, as index

selection cannot be expressed in terms of min only. However,

the property max{ak − bj} = max{ak} − min{bj} for any

two collections of numbers {ak} and {bj} proves useful

in reducing the proof to the already-studied min case. An

important property of the max-min arrangement is that there

exists a finite time after which switching is restricted to

happen either among the min or the max selectors, whichever

group contains the terminal index. Indeed, as in the min and

max cases, there is a unique terminal regulator index corre-

sponding to a unique equilibrium point, as seen next. The

stability proof for this case requires more space than alloted

in this paper. It is contained in an upcoming publication by

this author. For the remainder of the paper, it is assumed that

the terminal regulator index belongs to the min set, that is,

i∗ ∈ L.

Proposition 4: Under the max-min switching law of Eq. 1,

system (2, 3) has a unique equilibrium point at (xi∗ , ui∗),where i∗ ∈ L ∪ H is the index satisfying condition 26:

0 ≥ −sign (∆k,i∗)

Θk

∀k ∈ H (26)

and either condition 27 or condition 28:

0 ≤ −sign (∆j,i∗)

Θj

∀j ∈ L (27)

0 > minj∈L

{

−sign (∆j,i∗)

Θj

}

(28)

When condition 27 is satisfied, the terminal regulator index

i∗ ∈ L. Otherwise, conditions 28 is satisfied and i∗ ∈ H .

The details of the stability proof for combined max-min

switching are omitted due to space limitations.

V. INVARIANCE PROPERTIES: LIMIT PROTECTION

In this section, we show that the min, max and max-

min designs actually maintain outputs within certain limits.

When the min switching law is used alone, outputs whose

Θ is positive will be protected against upper-limit violations

and outputs whose Θ is negative will be protected against

lower-limit violations. Conversely, the max switching law

alone protects outputs whose Θ is positive against lower-

limit violations and outputs whose Θ is negative against

upper-limit violations. A max-min scheme is used to cover

additional combinations of signs of Θ and upper or lower

limits. An interval (−∞, b] is invariant for a generic real

variable z(t) if z(t) ≤ 0 at z = b. Similarly, an interval

[a,∞) is invariant if z(t) ≥ 0 at z = a. When an interval is

invariant and z(t1) belongs to the interval for some t1 > 0,

then z(t) will remain in the interval for t ≥ t1. For the



158

proposed technique to be effective, the interval (−∞, 0]must be invariant for upper-limited variables, in view of the

definition of sj for limited output yj . Conversely, [0,∞)must be invariant for lower-limited variables.

A. Invariance under Min Switching

Let sj be limited variable. The derivative of sj when i is

active is given by Eq. 16. When i is active we must have

ui ≤ uj , so:

sj|i

Θj

= Γj,ixa −ηi

Θi

sign (si) ≤ −ηj

Θj

sign (sj)

Noting that the inequality changes to equality for j = i, it is

clear thatsj

Θj≤ 0 at sj = 0 under any regulator. If Θj > 0,

upper-limit protection is guaranteed. If Θj < 0, lower-limit

protection is guaranteed.

B. Invariance under Max Switching

Following the same reasoning used for the min case, it is

clear thatsj

Θj≥ 0 at sj = 0 under any regulator. If Θj > 0,

lower-limit protection is guaranteed. If Θj < 0, upper-limit

protection is guaranteed.

C. Invariance under Max-Min Switching

Although intuition would indicate that the max-min ar-

rangement always protects upper and lower-limited variables

regardless of the signs of their respective Θ, it is not the

case. An exception occurs for sj when j ∈ L and the active

regulator belongs to H . This lack of symmetry arises from

the fact that for q ∈ H to be active it is necessary that

urq be greater than the minimum of all url, l ∈ L, and not

for every url in L. By contrast, for q ∈ L to be active, url

must be greater than every urh, h ∈ H . Fortunately, separate

arguments can be made which maintain the validity of the

approach under commonly-found circumstances. Details are

omitted due to space limitations, but will be included in an

upcoming publication by this author.

VI. DESIGN PROCESS

This paper deals with a single control input and a single

controlled output, whose setpoint is to be changed. Of all

outputs, exactly one is the controlled variable, while the

rest are limited variables. The Gi and Θi for the limited

outputs are given by the system definition and are thus not

design freedoms. The fundamental assumption that they de-

fine minimum-phase outputs must hold for this technique to

work, however. The designer may freely choose all switching

gains ηi and the reference states xi and ui so that they

constitute equilibrium pairs and so that they correspond to

the desired setpoint for the controlled variable and to the

limits yi. Next, upper-limited variables s.t. Θi > 0 and

lower-limited variables s.t. Θi < 0 are placed under the

min selector, while upper-limited variables s.t. Θi < 0 and

lower-limited variables s.t. Θi > 0 are placed under the max

selector. For the purposes of showing stability, no distinction

was made between regulated and limited outputs, and the

sliding function for the regulated variable was defined as

the difference between the output and a limit. Suppose y0 is

the true system output to be regulated. Since output setpoint

regulation is equivalent to the selection of a reference pair

(x, u), one may introduce a limited variable and associated

regulator for the purpose of reaching the reference state.

Without loss of generality, suppose that y1 = G1x + Θ1u

is such variable. For y0 to reach its setpoint, (x1, u1) must

be chosen under the restriction that G1x + Θ1u equals the

desired setpoint, and G1 and Θ1 must be chosen to satisfy

Assumption 2 and for good performance during the sliding

mode. The regulator for y1 is then placed under the min

selector. Under nominal conditions, the designer ensures that

i∗ = 1, so that y1 attains the commanded setpoint. This is

easily accomplished by manipulation of Θ1, η1 since the

choices do not affect the ability to place the eigenvalues of

Aeq,1, nor compromise system stability.

VII. EXAMPLE

Consider the pair (A, B) with

A =

1 0 −1−1 1 −1−1 −2 −1

and B = [1 0 1]T . This system is open-loop unstable. Since

it has no poles at zero, Assumption 1 is satisfied. Suppose

that the regulated output is y0 and there are 3 limited outputs,

defined as in Eq. 4 with

G2 = [−55 − 250 115] , Θ2 = 10

G3 = [23.5 120.5 − 57.5] , Θ3 = −4

G4 = [−2 − 20 9] , Θ4 = 1

It can be verified that these outputs are minimum-phase,

satisfying Assumption 2. The sliding function coefficients

for s1 are chosen so that Aeq,1 has desired eigenvalues.

For instance the eigenvalues -4 and −2 ± 2i are obtained

with G1 = [22.5 77.5 − 40.5] and Θ1 = −2. Let y2

and y3 be upper-limited at 16 and 8, respectively, and y4

lower-limited at -1. Since Θ2 > 0, the output limit regulator

#2 is applied to the min selector. Similarly, since Θ3 < 0and Θ4 > 0, regulators #2 and #3 are applied to the max

selector. Regulator #1 is associated with the min selector,

since it corresponds to the regulated output. The reference

states for y2, y3 and y4 are found by simultaneously solving

yj = Lj and x = 0 for xj and uj for j = 2, 3, 4, where

Lj are the desired limits. Let the setpoint of y0 be such that

x = [0.25 −0.25 −0.5] and u = −0.75, which corresponds

to a setpoint of 8 for y1. Under nominal conditions, one

wishes that #1 be the terminal regulator. Tuning is then

performed using the switching gains ηi until the conditions

of Proposition 4 hold for i∗ = 1. It can be verified that the

values η1 = 5, η2 = 1, η3 = 20 and η4 = 15 are such that

i∗ = 1 and i0 = 1, assuming zero initial conditions for the

determination of the latter. Figure 2 shows how the system

operates at the design point. It can be observed that y1 attains

its setpoint and that y3 and y4 ‘ride’ their limits during the

transient regime. Figure 3 illustrates system behavior under



159

off-nominal conditions, namely an increase in the setpoint of

y1 to 10. Under these conditions, the steady value of y3 is

incompatible with i∗ = 1 and the limit of 8. For this reason,

regulator #3 becomes active at steady-state, and it is natural

that y1 does not reach its setpoint. Similar protective behavior

-transient or steady- may be observed upon re-tuning of η or

G. It can be determined that, all other parameters equal, the

setpoint for y1 may be changed in the range [−7.87, 8.53]without a change in i∗, partly defining ‘nominality’ for this

example. The cartesian product of the admissible intervals

for all adjustable parameters (including G1 and Θ1, all η

and all yi) so that i∗ = 1 completely defines nominality. It

is a separate problem to estimate this region of nominality

given A, B and the G and Θ that correspond to the limited

outputs. Figure 4 shows the switching history for the nominal

and off-nominal cases. As it can be seen, there is a time

beyond which the switchings occur only within the selector

containing i∗, the min selector in this case.

VIII. CONCLUSIONS AND FUTURE WORK

A multi-sliding mode regulator scheme with max-min

selection logic was proposed. The stability of the min-max

arrangement is guaranteed when the individual regulators

are designed separately, following the standard sliding mode

design procedure. It was established that the min selector

protects upper-limited outputs whose direct term (’D’ matrix)

is positive and lower-limited outputs whose direct term is

negative. The max selector offers the opposite protections.

When used in combination, the max and min selectors

offer additional protection. Further research is necessary to

improve the design process, mainly to determine a procedure

by which any given regulator can be made to be the terminal

one. Immediate extensions of the work include the inclusion

of matched disturbance and the development of a multi-input

version of the technique.

REFERENCES

[1] K.J. Astrom and T. Hagglund. PID Controllers: Theory, Design and

Tuning. Instrument Society of America, 1995.[2] A. Bemporad, F. Borrelli, and M. Morari. Model predictive control

based on linear programming - the explicit solution. IEEE Transac-tions on Automatic Control, 47(12):1974–1985, 2002.

[3] A. Bemporad et al. The explicit quadratic regulator for constrainedsystems. Automatica, 38(1):3–20, 2002.

[4] F. Blanchini. Set invariance in control. Automatica, 35:1747–1767,1999.

[5] M.S. Branicky. Analysis of continuous switching systems: Theory andexamples. In Proc. American Control Conference, pages 3110–3114,1994.

[6] M.S. Branicky. Multiple Lyapunov functions and other analysis toolsfor switched and hybrid systems. IEEE Transactions on Automatic

Control, 43(4):475–482, 1998.[7] A.M. Foss. A criterion to assess stability of a lowest-wins control

strategy. IEE Proceedings, Part D, 128(1):1–8, 1981.[8] M. Johansson. Piecewise linear control systems. In Lecture Notes in

Control and Information Sciences, volume 284. Springer, 2003.[9] J.S. Litt et al. The case for intelligent propulsion control for fast

engine response. In AIAA Infotech@Aerospace Conference, AIAApaper AIAA-2009-1876, 2009.

[10] H. Richter, B.D. O’Dell, and E.A. Misawa. Robust positively invariantcylinders in constrained variable structure control. IEEE Transactions

on Automatic Control, 52(11):2058–2069, 2007.

[11] H. Richter, A.V. Singaraju, and J.S. Litt. Multiplexed predictive controlof a large commercial turbofan engine. AIAA Journal of Guidance,

Dynamics and Control, 31(2):273–281, 2008.[12] H.A. Spang and H. Brown. Control of jet engines. Control Engineering

Practice, 7:1043–1059, 1999.[13] J. Zhao and D.J. Hill. On stability, L2-gain and H∞ control for

switched systems. Automatica, 44(5):1220–1232, 2008.

0 2 4 6 80

2

4

6

8

10

t

y1

Regulated Output

0 2 4 6 8−20

−15

−10

−5

0

t

y2

Upper−limited Output

0 2 4 6 80

2

4

6

8

10

t

y3


0 2 4 6 8−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

t

y4

Lower−limited Output

Active LimitActive Limit

Active Limit

Inactive Limit

Fig. 2. System Behavior Under Nominal Conditions

0 2 4 6 80

2

4

6

8

10

12

t

y1

Regulated Output

0 2 4 6 8−20

−15

−10

−5

0

t

y2


0 2 4 6 80

2

4

6

8

10

t

y3


0 2 4 6 8−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

t

y4

Lower−limited Output

Inactive Setpoint

Active Limit

Active Limit

Inactive Limit

Fig. 3. System Behavior Under Off-Nominal Conditions

0 1 2 3 4 5 6 7 80

1

2

3

4

5Selection History: Nominal Conditions

q(t

)

t

0 1 2 3 4 5 6 7 80

1

2

3

4

5

q(t

)

t

Selection History: Off−Nominal Conditions

Min set becomes invariant

Fig. 4. Switching Histories Under Nominal and Off-Nominal Conditions



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