Toward unified analysis and controller synthesis for a class of hybrid systems

Nonlinear Analysis 65 (2006) 2216–2235www.elsevier.com/locate/na

Toward unified analysis and controller synthesis for aclass of hybrid systems

Luis Rodriguesa,∗, Jonathan P. Howb

a Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canadab Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Boston, MA, USA

Abstract

This work defines a new class of hybrid systems called state-based switched (SBS) systems that havenumerous important engineering applications. The characterizing feature of these systems is that thediscrete-event dynamics are associated with the continuous-time state making a specific function be equalto zero. The choice of this function is application specific and for the closed-loop SBS systems definedin this paper it is related to the execution of a desired set of tasks from a pre-specified mission plan. Forthis broad class of SBS systems, the paper presents a unified analysis and controller synthesis methodologybased on Lyapunov theory. Depending on the details of the mission plan, the closed-loop hybrid systemwill be divided into two subclasses: sequential and non-sequential. The controller design procedure forboth subclasses consists of the same two steps: finding a control law and finding a stabilizing switchingrule. For static state and output feedback of sequential hybrid systems, the paper proposes a new hybridsequential sliding-mode controller. It is proven that the control mission can be accomplished for sequentialhybrid systems under static state and output feedback using this new controller. A similar framework isinvestigated for the more complex class of nonsequential hybrid systems and a systematic procedure fordesigning the switching rule is presented for some specific instances of these systems.c© 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Hybrid systems, which incorporate both discrete event-driven and time-driven dynamics thatinteract at the event times, have been a very active area of research over the past decade [1–8].

∗ Corresponding author. Tel.: +1 514 8482424x3135; fax: +1 514 8483175.E-mail addresses: [email protected] (L. Rodrigues), [email protected] (J.P. How).

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http://dx.doi.org/10.1016/j.na.2006.02.048

L. Rodrigues, J.P. How / Nonlinear Analysis 65 (2006) 2216–2235 2217

There are many classes of hybrid systems, including piecewise-affine systems [9–13], logic-based control systems [14], switched systems [14–17] and sliding modes [18]. Applicationareas include computer disk drives [19], automotive systems [20–22], aerospace systems [23,24], manufacturing systems [25] and many others. For a review of the research done on hybridsystems up to 2001, see the special issues on hybrid systems [1,2,4–6,8]. Much of the previousresearch on hybrid systems has been focused on modeling and simulation [16,26–37] and ondeveloping stability and reachability analysis methods [3,9–11,38–43]. Some synthesis methodshave also been developed [10,14,44–48], but these mainly focused on state feedback and/or havebeen limited to specific applications and discrete-time approximations of continuous dynamics.There have been some important research efforts toward an overall unified model [49–51] and aunified analysis methodology for a class of hybrid systems [49,52], as well as a unifying view forthe subclass of piecewise-affine systems [53]. However, there have been very few contributionstoward a unified controller synthesis method that would provide a systematic control design toolfor a large class of hybrid systems and enable designers to use the same methodology for a broadset of models and applications. With this goal in mind, the current paper presents a first steptoward a unified controller synthesis methodology for a particularly important class of hybridsystems: state-based switched (SBS) systems. The characterizing feature of these systems isthat the discrete-event dynamics are associated with the continuous-time state making a specificfunction be equal to zero. The choice of this function is application specific and for the closed-loop SBS systems defined in this paper it is related to the execution of a desired set of tasks froma pre-specified mission plan.

Previous work of the authors [12,13] developed synthesis methods for piecewise-affine (PWA)systems, which are a special class of hybrid systems in that the continuous-time dynamicswithin each discrete mode are affine and the switching events are associated with discrete modeswitching. Furthermore, this switching always occurs at very specific subsets of the continuous-time state space that are known a priori. This paper extends the previous results on PWA systemsto synthesize controllers for the more general class of SBS systems for which the event timesare determined by the continuous state making a specific function be equal to zero. The primarydifference from PWA systems is that the dynamics in each mode of the SBS systems are notnecessarily affine. Instead, the dynamics in each SBS system mode can be nonlinear and canbe associated with different tasks to be performed by the system rather than being associatedwith different affine models. Important examples of SBS systems are task-level control systemsfor which a discrete mode is associated with each task to be performed. For example, as willbe shown in the paper, this approach can be used to design controllers to perform complexmaneuvers (e.g., parking) for autonomous vehicles obtained by patching together basic primitivemaneuvers, each maneuver being a task. Another example of a SBS system addressed in thispaper is car driving where each mode corresponds to the active gear and each gear is switchedbased on the velocity of the car (a continuous-time variable).

Building on previous research on piecewise-affine and hybrid systems [10,12,13,53,16,40], multiple Lyapunov functions will be used to develop a unified analysis and synthesismethodology for hybrid systems that designs one Lyapunov function for each task to beperformed by the SBS system. In contrast to previous research on piecewise-affine systems [10,12,13,53], it is not possible to use different quadratic sectors of the same Lyapunov function,but multiple quadratic Lyapunov functions can be considered instead. This extends the work onpiecewise-quadratic Lyapunov functions [53] and the work on using multiple Lyapunov functionsto design switching rules that stabilize a given set of vector fields [14,15] to consider the design ofthe vector fields themselves. The main result of this paper is important because it is typically not

2218 L. Rodrigues, J.P. How / Nonlinear Analysis 65 (2006) 2216–2235

clear how to design the closed-loop vector fields for a given application and because it formulatesthe synthesis problem at the task-level control of hybrid systems having mission execution plansfor the tasks. This paper addresses these issues by developing a systematic Lyapunov-basedmethodology to synthesize the closed-loop vector fields and a systematic approach to the designof the switching rule for the tasks involved in the mission plan of a subclass of SBS systemscalled sequential hybrid systems.

Based on the observations of the previous paragraphs, there are four main objectives andcontributions of this paper:

1. To define a new class of hybrid systems called SBS systems, and illustrate thatthis encompasses numerous important engineering applications, including instances ofunderactuated systems and systems with nonholonomic constraints;

2. To define unified analysis and control design problems for SBS systems;3. To further classify closed-loop SBS systems into two subclasses for design purposes:

sequential and nonsequential systems. For static state and output feedback of sequential hybridsystems the paper proposes a new hybrid sequential sliding-mode controller;

4. To use multiple Lyapunov functions for solving the unified controller synthesis problemfor SBS systems and to give a formal proof that the control mission can be accomplishedfor sequential hybrid systems under static state and output feedback using the proposedmethodology.

The paper starts by defining the class of hybrid systems. This is followed by the unifiedLyapunov-based analysis and synthesis problems. Next, the formal solution of the control designproblem for static state and output feedback of sequential hybrid systems is presented. Finally,several examples are presented to show the effectiveness and applicability of the proposedmethodology to both sequential and nonsequential hybrid systems.

2. Class of hybrid systems

The hybrid systems considered in this work are state-based switched systems. This class ofsystems consists of the following components:

• State space — The state of the system consists of a discrete-event parameter σp(t) ∈{1, . . . , N} and a continuous-time state vector x(t) ∈ R

n .• Continuous-time dynamics — The continuous-time dynamics are governed by the differential

equation

x(t) = fσp(t)(x(t))+ gσp(t)(x(t)) · u(t), (1)

y(t) = Cσp(t)x(t) (2)

where u ∈ Rm is the control input and y ∈ R

p is the output.• System trajectories — Trajectories of system (1) will be considered in the sense of

Filippov [55], i.e., the solution will be an absolutely continuous function x(t) such that thereexists an index set I(t) and scalars λi (t) ≥ 0, i ∈ I(t), ∑i∈I(t) λi (t) = 1 such that

x(t) =∑

i∈I(t)λi (t)( fi (x(t))+ gi (x(t))u(t)), (3)

or, equivalently, x ∈ Coi∈I(t) { fi (x)+ gi (x)u(t)}, where the operator Co denotes the convexhull of the vector fields fσp(t)(x(t)) + gσp(t)(x(t))u(t) that are active at a given time t ≥ 0.Expression (3) is called a differential inclusion.


• Switching rule — The switching of the discrete parameter σp from i to j occurs when a givenfunction of the state hpi j (x) is equal to zero.

• Performance output — There exists a performance output of the plant s = Cx , s ∈ Rl .

• Error signal — Associated with the performance output there is an affine change ofcoordinates providing an error signal e(s) = Ws + n.

Note that discrete inputs or outputs are not allowed for systems in this class. Furthermore, anysystem in this class must also be linear in the continuous-time input and must have an outputthat is linear in the state. The next two examples show how switching in thermal systems can beincorporated into the class of systems defined in this paper in two different ways: by having atemperature-dependent coefficient or by switching an external fan on or off.

Example 2.1 (Heat Exchanger with Temperature-dependent Coefficient). This exampleconsiders a heat exchanger system connected to a heat source with unitary flow rate q0 andwhose temperature1 obeys the first order differential equation

T = c(T )T + q0. (4)

The temperature coefficient has the following piecewise-constant characteristic

c(T ) ={

1 T < 0,−1 T > 0,

which (setting x = T ) divides the state space into the polytopic regions

R1 = {x ∈ R | x < 0}, R2 = {x ∈ R | x > 0}.If σp is used as a region indicator (σp = 1 for x ∈ R1 and σp = 2 for x ∈ R2), the dynamicswill then be described by the piecewise-affine system

x = Aσp x + bσp

with A1 = 1, A2 = −1, b1 = b2 = 1. The dynamics described by the pair (A1, b1) arevalid for x ∈ R1 and the dynamics described by (A2, b2) are valid for x ∈ R2. In this case,the performance output and the output are the same as the state, i.e., the temperature. Given areference temperature Tref, the error signal is simply e(x) = x − Tref. This system belongs to theclass (1) and (2) with hp12(x) = hp21(x) = x ,

f1 = A1x + b1, g1 = 0, C1 = C = W = 1, n = −Tref

f2 = A2x + b2, g2 = 0, C2 = C = W = 1. �

Example 2.2 (Computer-Fan Cooling System). A computer has a fan that is turned on to cool itdown. The electronic equipment has a thermal capacitance Cthermal and generates heat at theconstant rate q0 when it is operating. The equipment’s thermal conductivity to the ambienttemperature due to convection is η with the fan off and 2η with the fan on. The ambienttemperature is Ta , and the temperature of the electronic equipment is T . Let σp(t) ∈ {1, 2}be associated with the fan switching on (σp = 1) and off (σp = 2). The dynamics of this systemare then given by

T = cσp (−T + Ta)+ 1

Cthermalq0, (5)

1 The temperature is expressed in degrees Celsius.


where

cσp =

⎧⎪⎪⎨⎪⎪⎩

2η

Cthermalσp = 1,

η

Cthermalσp = 2.

In this case, the performance output and the output are the same as the state, i.e., the temperaturex = T . Given a reference temperature Tref, the error signal is simply e(x) = x − Tref. If theswitching on and off of the fan depends on the state x such that

σp(t) ={

1 x(t) ≥ Tref,

2 x(t) < Tref,

then the resulting system belongs to the class (1) and (2) with hp12(x) = hp21(x) = e(x) =x − Tref and

f1 = 2η

Cthermal(−x + Ta)+ 1

Cthermalq0, g1 = 0, C1 = C = W = 1, n = −Tref

f2 = η


Cthermalq0, g2 = 0, C2 = C = W = 1. �

The next example shows how another hybrid system with a discrete-event input can betransformed into the framework of this paper when the discrete-event input is designed to dependon the continuous-time state.

Example 2.3 (Gear Shifting in a Car). Consider a simplified model of a car with two gears. Thecar moves in 1D with position coordinate z and the continuous state is x = [z v]T , where v = zand 0 ≤ v(t) ≤ vmax. The dynamical equations are

x1 = x2,

x2 = wσp(t)(x2)u,

y = x2,

where u is the continuous-time input (scaled acceleration) and σp(t) ∈ {1, 2} is the gear number,a discrete-event input. The functions wσp (x2) take values in the interval [0, 1] and model theefficiency of each gear. Typically,w1(x2) is a continuously decreasing affine function of x2 withw1(0) = 1, w1(vmax) = 0 and w2(x2) is an continuously increasing affine function of x2 withw2(0) = 0, w2(vmax) = 1, which implies that

∇12(x2) = ∂w1

∂x2(x2)− ∂w2

∂x2(x2) < 0, ∀x2. (6)

The performance output of the system is the velocity and therefore C = [0 1]. The objective isto use the efficiency of the gears and switch them such that the maximum velocity is achieved.Therefore, e(x) = x2 − vmax, W = 1, n = −vmax. If the gear shifting function is designed todepend on the continuous-time state as

σp(t) ={

1 w1(x2(t)) ≥ w2(x2(t)),2 w1(x2(t)) < w2(x2(t)),


then the resulting system also falls under the framework of this paper with hp12(x) = hp21(x) =w1(x2)−w2(x2) and

f1 = [x2 0]T , g1 = [0 w1(x2)]T , C1 = C = [0 1], W = 1, n = −vmax

f2 = [x2 0]T , g2 = [0 w2(x2)]T , C2 = C = [0 1]. �

Having defined the class of hybrid systems that will be the focus of the paper we nowpropose a Lyapunov-based methodology for the solution of the analysis and controller synthesisproblems.

3. Unified analysis and controller synthesis

This section starts by analyzing the error trajectories of system (1) and (2) when the input hasbeen replaced by a given known function of the state (if the zero function is used the open-loop trajectories are obtained). This is followed by a Lyapunov-based method to synthesizecontrollers for a specified mission using multiple Lyapunov functions. The synthesis is doneas a refined analysis of the closed-loop system where the controller parameters are involved.As is typically the case with Lyapunov-based methods, the proposed method is not guaranteedto work for all systems. However, when it works the result is unambiguous: both a controllerand a Lyapunov function that proves stability of the closed-loop system can be found. Severalexamples are presented to show the effectiveness of the method in Section 5.

3.1. Unified analysis

The stability of the error trajectories of system (1) and (2) can be analyzed using a candidateLyapunov function of the error trajectories

V (e), e = WCx + n. (7)

We suggest that V (e) be chosen to be a polynomial and therefore V (e) is a smooth function.

Remark 1. If the squared error (a quadratic polynomial) is chosen as the candidate Lyapunovfunction it will be of the form

V (x) = 1

2eT e = 1

2

(x T Px + 2qT x + r

), (8)

where P = CT W T WC, q = CT W T n, r = nT n. �

Note that the globally quadratic candidate Lyapunov function (8) is positive definite and verifiesthe constraint V (x) = 0 for all x such that e(x) ≡ WCx + n = 0, which guarantees thatthe error signal e will be zero when V (x) = 0. In fact, the quadratic Lyapunov function isa perfect square of the norm of the error and therefore it is positive definite. However, wewill not restrict the analysis to quadratic candidate Lyapunov functions or to polynomials thatare perfect squares. Therefore, a means to determine positive definiteness of a polynomial isneeded. Unfortunately, verifying that a given polynomial is non-negative is in general an N Phard problem [56]. Following [57] this condition will be relaxed to ask that the polynomial beinstead a sum of squares (SOS) [58].


Definition 3.1 ([57,58]). For e ∈ Rn a multivariate polynomial V (e) is a sum of squares (SOS)

if there exist some polynomials Vi (e), i = 1, . . . ,M , such that

V (e) =M∑

i=1

V 2i (e). (9)

A sum of squares is only positive semi-definite. To guarantee positive definiteness, the followingresult can be used.

Lemma 3.1 ([58]). Given a polynomial V (e) of degree 2d, let

φ(e) =n∑

i=1

d∑j=1

εi j e2 ji (10)

d∑j=1

εi j > γ, ∀i = 1, . . . , n, (11)

where γ > 0, εi j ≥ 0, ∀i, j . Then the condition V (e) − φ(e) is SOS guarantees the positivedefiniteness of V (e).

Proof ([58]). The function φ(e) defined above is obviously positive definite if all εi j verify theconditions in the lemma. Then V (e)− φ(e) being SOS implies that V (e) ≥ φ(e), which in turnimplies the positive definiteness of V (e). �

The following result now establishes sufficient conditions for the stability of the error trajectoriesof the hybrid system.

Theorem 3.1. If the candidate Lyapunov function (7) verifies V (e) = 0 if and only if e = 0,

dV

dt= dV

de

de

dxx < −αV , (12)

for α > 0, where x represents the system dynamics given by (3) when the input has been replacedby a given known function of the state (if u = 0 is used, the open-loop trajectories are obtained),and either of the conditions

1. V (e) is a perfect square or, more generally,2. There exists φ(e) verifying (10) and (11) such that V (e)− φ(e) is SOS

then V (e) becomes a Lyapunov function and the error trajectories converge exponentially tozero.

Proof. The result follows from standard Lyapunov theory [59]. In fact, V (e) is positive definitebecause either V (e)−φ(e) is SOS with φ(e) verifying (10) and (11), or V (e) is a perfect square,and V (e) = 0 if and only if e = 0. Moreover, condition (12) implies V (t) ≤ V (0)e−αt forα > 0 and thus V (e) converges to zero exponentially because α > 0. Since V (e) = 0 if and onlyif e = 0 and, moreover, V (e) can be lower bounded by a function of ‖e‖2 then e(t) convergesexponentially to zero. �

Remark 2. For a detailed explanation on how SOS Lyapunov functions can be computednumerically using the toolbox SOSTOOLS please refer to [58,60]. �


3.2. Unified synthesis

When the error trajectories of a system in the class (1) and (2) are not stable, a controller mustbe designed to stabilize the system trajectories. In this section, a generalization of the stabilizationproblem will be defined whereby a system in the class (1) and (2) will be controlled to performa set of tasks in a mission execution plan. Even when the controller parameters are known, theproblem formulated in this section is more general than the one formulated in Section 3.1 in thatintermediate tasks can also be defined. Based on these considerations, for a plant in the classof systems (1) and (2), it will be assumed that the control objective is described by a missionexecution plan with M ≥ N tasks to be performed. More specifically, the mission and controlshould have the following properties:

• Discrete state — One controller should be designed for each of the M tasks and the controllerswill be indexed by the discrete parameter σc ∈ {1, . . . ,M}.

• Continuous-time state — The controller can either provide static or dynamic feedback. In thelatter case, the controller state is xc ∈ R

n .• Objective function — The control objective will be based on a performance output that can be

different for different controllers. More specifically, s = Cσc x , s ∈ Rl . Associated with the

performance output and for each value of the discrete parameter σc there is an affine objectivefunction (an error signal) eσc(s) = Wσc s+nσc which should be driven to zero by the controllerdesigned for the discrete mode σc, meaning that the corresponding task has been performed.

• Controller switching — The controller design includes the synthesis of a switching rule thatswitches among controllers to make sure that the requirements of the mission execution planare met. The controller switching verifies a similar constraint to the plant switching, namely,there must exist a function hci j (y, s, xc) which is equal to zero at a controller switch fromσc = i to σc = j .

Remark 3. The controller switching property enables different switching times for the plant andthe controllers. In the particular case of static state and output feedback, the control is a staticfunction of the state or output and, therefore, there is no controller state xc. Moreover, when thestate of the system is available, y = x . �

Remark 4. Note that the objective function for the controller has added more flexibility tothe closed-loop system as compared to the open-loop system. Namely, in closed-loop there isone error function for each controller of the hybrid system. The error dynamics will thereforechange (switch) as the controller changes (switches). However, if the designer so wishes, thetasks associated with different controllers can be defined to be the same and the structure of theopen-loop system is recovered. �Depending on the mission requirements, closed-loop hybrid systems will be divided into twosubclasses. Before defining these subclasses the following concept is needed.

Definition 3.2. A mission plan is a finite list of tasks.

Using the concept of mission plan, sequential hybrid systems can now be defined.

Definition 3.3. A Sequential Hybrid System (SHS) is a system for which the tasks specified inthe mission plan must be executed sequentially with a pre-determined ordering. The tasks can beexecuted more than once if the mission plan is repeated. In this case, it is said that there existsmore than one run of the mission plan. More specifically, the following requirements should bemet:


1. σc is defined to be the ordering number i of the tasks.2. A task i is finished, or performed, when ei (s) = Wi s + ni = 0.3. Each task must be performed in finite time, except possibly for the last task, if the mission

plan is not repeated. If the mission plan is repeated and consists of more than one run, at theend of each run through the task list, it must be ensured that eM (s) = 0.

4. The tasks must be performed sequentially. Therefore, there can only occur a switch fromσc = i to σc = i + 1. This switch occurs at switching time ti , which is the time for whichei (s) = Wi s + ni = 0. Therefore, for SHS we have j = i + 1 and hci j (y, s, xc) = ei (s).The control signal active for a given value of σc = i must therefore be designed to makeei (s) = Wi s + ni = 0 in finite time, except possibly for the case σc = M , where the objectivecan be achieved asymptotically if the mission consists of only one run through the task list.

5. After a switch from i to i + 1 at ti , the value of σc is not allowed to be equal to i again unlessthe mission includes more than one run. In that case, σc is only allowed to be equal to i afterit has gone through an entire loop until σc = i − 1 again, where i − 1 ≡ M for i = 1.

Definition 3.4. A Non-Sequential Hybrid System (NSHS) is a system which is not sequentialaccording to Definition 3.3. The control signal active for a given value of σc = i must be designedto make ei (s) = Wi s + ni = 0. In this case, however, there is no particular ordering of the tasksbut there is the requirement that at the end of the mission when t → t f (possibly t f = ∞),e1(s) = · · · = eM (s) = 0. More specifically, the switching rule must be provably stabilizing inthe sense that it must ensure that e1(s) = · · · = eM (s) = 0 when t → t f .

Depending on the strategy adopted to control a piecewise-affine system and on the systemexecution, a closed-loop piecewise-affine system can fall under the framework of either an SHSor an NSHS, as shown in the next two examples.

Example 3.1 (Piecewise-affine Systems as Sequential Systems). This example continues theanalysis of the heat exchanger system connected to a heat source with unitary flow rate q0 andinput flow rate u(t). The temperature now obeys the forced first order differential equation

T = c(T )T + q0 + u,

where, as before,

c(T ) ={

1 T < 0,−1 T > 0,

which (setting x = T ) partitions the state space into the polytopic regions

R1 = {x ∈ R | x < 0}, R2 = {x ∈ R | x > 0}.If the index i is used as a region indicator, the dynamics will then be described by the piecewise-affine system

x = Ai x + bi + Bi u

with A1 = 1, A2 = −1, b1 = b2 = 1, B1 = B2 = 1. The dynamics described by thetriple (A1, B1, b1) are valid for x ∈ R1 and the dynamics described by (A2, B2, b2) arevalid for x ∈ R2. Suppose that a state feedback controller must be designed to globallyasymptotically stabilize the open-loop unstable equilibrium point of region R1, verifying theequation A1x + b1 = 0 (i.e., x = −1). Assume that it is known that the initial temperature isalways positive (i.e., it is located in region R2) so that we are interested in executions with theinitial state x0 ∈ R2. Since the system to be controlled is piecewise-affine, a piecewise-affine


controller will be designed. Since the number of regions is N = 2 in this case, there will beone controller for each region and, consequently, M = 2. A discrete parameter σ is associatedwith each of the different operating regions. This parameter will be the same for the plant andthe controller because a piecewise-affine system switches based on the state, which in this caseis the temperature, by assumption, also available to the controller. Let the task of the controllerfor x ∈ R2 (σ = 2) be to take the system to the boundary between regions R1 and R2 describedby the equation

e2(x) = W2C2x + n2 = 0 (13)

with W2 = 1, C2 = 1, n2 = 0. The task of the controller for σ = 1 is to take the system to theclosed-loop equilibrium point x = −1 which verifies

e1(x) = W1C1x + n1 = 0 (14)

with W1 = A1,C1 = 1, n1 = b1. Since the two tasks must be executed sequentially whenthe initial state x0 ∈ R2 (from x0 ∈ R2, go first to the boundary with R1, and then go from theboundary to the equilibrium point inside R1), this is an example of a sequential hybrid system forall executions where x0 ∈ R2. For this particular example, the switching function is the same forthe controller and the plant hc12(x) = hc21(x) = hp12(x) = hp21(x) = e2(x) = x . The controlstrategy for piecewise-affine systems given above is similar to the one proposed in [61]. �

Example 3.2 (Piecewise-affine Systems as Non-Sequential Systems). Consider the same systemas in Example 3.1 with the same control objective. Assume the temperature is measured, butthe measurement is noisy. It is then necessary to estimate the temperature to determine when toswitch controllers. Therefore, the strategy now is to use a dynamic output feedback controllerdesigned to stabilize the open-loop equilibrium point of R1. The switching function of the plantis again hp12 = hp21 = x . However, since the controller now has its own state xc, a switchingfunction is also needed for the controller that will be of the form hc12 = hc21 = xc. Note that ingeneral xc(t) = x(t). Suppose now that the controllers from regions R1 and R2 have the sametask, which is to stabilize the system to the desired closed-loop equilibrium point x = −1, whichverifies e1(x) = 0 with e1 defined in (14). Because of the control strategy adopted, e2 = e1.Since there is no specific ordering for the tasks, but it is desired that both e1 → 0 and e2 → 0as t → ∞ (i.e., the equilibrium point is made asymptotically stable), this is an example of anon-sequential hybrid system. �

For a system described by (1) and (2) and for which a mission has been planned, the unifiedcontroller synthesis problem can now be defined.

Definition 3.5. For a system described by (1) and (2), the controller synthesis problem is thefollowing: find a set of control signals ui , i = 1, . . . ,M , and a set of switching functionshci j (y, s, xc), i, j = 1, . . . ,M , such that:

• For sequential hybrid systems. All control mission objectives from Definition 3.3 are met inclosed-loop when the control signals ui are switched according to the switching functionshci j .

• For nonsequential hybrid systems. All control mission objectives from Definition 3.4 are metin closed-loop when the control signals ui are switched according to the switching functionshci j . �

The control signals will be found for this problem using a control Lyapunov function.


Remark 5. Based on the quadratic polynomial function (8) suggested in Remark 1, thefollowing candidate control Lyapunov function can be considered for each σ

Vσ (x) = 1

2

(x T Pσ x + 2qT

σ x + rσ), (15)

where x = [xT x T

c

]Tand σ = [

σp σc]T

. �

Remark 6. For state feedback where x = x, σp = σc = σ , obvious candidate control Lyapunovfunctions in the class (15) are

Vσ (x) = 1

2eTσ eσ = 1

2

(x T Pσ x + 2qT

σ x + rσ), (16)

where Pσ = CTσ W T

σ WσCσ , qσ = CTσ W T

σ nσ and rσ = nTσ nσ for each σ . Note that the candidate

control Lyapunov functions (16) are positive definite and verify the constraint Vσ (x) = 0 for allx such that eσ (x) ≡ WσCσ x + nσ = 0, which guarantees that the error signal eσ will be zerowhen Vσ = 0. As stated in Definitions 3.3 and 3.4, eσ should be driven to zero in finite time forSHS (except possibly for eM ) and eσ should be driven to zero asymptotically for NSHS. �

We now formally define our proposed method for Lyapunov-based controller synthesis.

Definition 3.6. Given candidate control Lyapunov functions Vσ (x), the controller design methodis comprised of the following two steps:

1. Design of the control signal. The control signal u from (1) will be designed such that, for eachmode σ , the following conditions are verified{

Vσ (x) > 0,d

dtVσ (x) < −ασV γσ

σ (x),(17)

for all possible values of x in mode σ and for some constants ασ ≥ 0 and γσ ≥ 0, which willbe selected according to the specific applications to be considered.

2. Design of the switching functions. The switching functions hci j (y, s, xc) should be designedsuch that the controller switching from σc = i to σc = j occurs when hci j (y, s, xc) = 0 andthe switching strategy is provably stabilizing. �

Remark 7. There is no constraint on the continuity of the different Lyapunov functions at theswitching and thus discontinuous Lyapunov functions are allowed. �

Having defined the Lyapunov-based controller synthesis method, formal results proving thatthe outcome of the method is a controller verifying the conditions of Definition 3.5 must bedeveloped. This paper gives a first step toward that goal by presenting in the next section a formalresult for static state and output feedback of sequential hybrid systems (SHS). Given the diversityof NSHS, a systematic procedure for finding the two design objects (control signal and switchingrule) can only be devised for special cases of NSHS. Some of these special cases will be coveredin the examples of Section 5. One such special case is the class of piecewise-affine systems forwhich the methodology developed in [12] can be used (see Example 5.4). Another special caseis stabilization and trajectory following for autonomous vehicles (see [54] and Example 5.6).


4. Solution of the design problem for SHS

The controller synthesis problem from Definition 3.5 can be solved systematically for staticstate and output feedback of sequential hybrid systems with all tasks performed in finite time.We first note that, according to Definition 3.3, the objective of the controller for each value ofσ is to drive eσ to zero in finite time. We then observe that the set {x | eσ (x) = 0} will be aswitching manifold in the state space. Therefore, the objective of the controller is to make thatmanifold an attractor for the trajectories of the dynamical system. For each σ = i , the trajectoriescan be forced to converge in finite time to the manifold ei (x) = 0 using a control signal with astructure similar to the one used for sliding modes [18,62] where a quadratic Lyapunov functionof the error is used to design the controller. Based on these observations, and following therequirements of Definition 3.3, we now establish the following result consisting of a new hybridsequential sliding-mode controller design method.

Theorem 4.1. For static output feedback control (state feedback if s = x) of an SHS, if in eachrun of the mission plan the control Lyapunov function verifies (17) and

Vi = 1

2eT

i ei , i = 1, . . . ,M

hci(i+1) = ei , i = 1, . . . ,M − 1

hcM1 = eM ,

hci j = β, i = 1, . . . ,M − 1, j = i + 1

γi = 1

2i = 1, . . . ,M

αi > 0,

(18)

where β is an arbitrary nonzero constant and a given mode i = 1, . . . ,M, will be maintainedif no switching function equals zero, all the control objectives stated in the mission plan ofDefinition 3.3 will be met in finite time. If there is more than one run, every finite set of runsis also performed in finite time.

Proof. For convenience, and without loss of generality, let the time parameter be re-initializedat zero after each switching in the system. For any i = 1, . . . ,M , when the system is in modeσ = i and before a switching to σ = i + 1 can occur there are two possibilities:

1. The manifold ei (x) = 0 is reached for a given time ti , possibly infinite,2. The manifold ei (x) = 0 is never reached.

In this proof we will show that 2 cannot happen and 1 implies that ti is finite.

Proof that 2 cannot happen. Assume that the manifold ei (x) = 0 is never reached. In this case,‖ei (x(t))‖ > 0, ∀t ≥ 0. This implies that given any t� ≥ 0, d

dt ‖ei‖ cannot be negative for allt ≥ t� including the asymptotic case when t → ∞. However, notice that the second inequalityin (17) with γσ = 0.5 yields

Vi < −αi V12

i ⇐⇒ ‖ei‖ d

dt‖ei‖ < − αi√

2‖ei‖ ⇐⇒ d


2< 0, ∀t ≥ 0 (19)

which is a contradiction.

Proof that if 1 is verified then ti is finite. Let ti be the first time for which the manifold ‖ei‖ =0 is reached, i.e., ei (x(ti)) = 0, where ti can possibly be infinite. Note that because ‖ei (x)‖ is


an affine function of x , it is continuous and, for continuous trajectories of x , if ‖ei (x(0))‖ > 0[‖ei (x(0))‖ < 0], then we can not have ‖ei (x(t))‖ < 0 [‖ei (x(t))‖ > 0] for t < ti . Notethat the first inequality in (17) is automatically verified by the Lyapunov function (18). SinceVi = 0.5‖ei‖2, from (17) and (18), the calculations from (19) can be performed again for t < ti ,yielding

d


2, (20)

where the relation ‖ei (x)‖ = 0, t < ti has been used. Integrating (20) from 0 to ti and takinginto account that ei (x(ti )) = 0 we finally get the relation

ti <

√2‖ei (0)‖αi

, (21)

which shows that the switching manifold ei = 0 is reached in finite time. This result in turn,proves that, for each σ = i , the trajectories will converge to the manifold ei = 0 in a finitetime ti . Note that, when ei = 0, the discrete parameter σ switches instantaneously to i + 1, andei+1(0) is finite. In fact, ei = Wi Ci x + ni = 0 at the switching, which implies x is finite, whichthen makes ei+1(0) finite. The same reasoning used for task i proves that if task i is performedin finite time, then so is task i + 1. Since, in particular, for i = 1, e1 = 0 is reached in finite time,we can now proceed by induction on i . Thus, by induction, for each i ∈ {1, . . . ,M}, the tasks areperformed in finite time and the switching to the next value of i is done instantaneously, whichproves that the whole set of tasks is performed sequentially and in finite time, in accordance withDefinition 3.3. If there is more than one run of the mission plan, since hcM1 = eM , the samereasoning proves that each run is executed in finite time and, therefore, every finite set of runs isalso performed in finite time. �

In summary, the result of this theorem provides a new hybrid sequential sliding-mode controllerdesign method whereby both a static output feedback input and a switching rule are designed fortask-level control of SHS in finite time.

5. Examples

This section presents six examples. The first three examples show how the result ofTheorem 4.1 can be applied to SHS. The remaining three examples show how to apply thesynthesis method from Definition 3.6 to specific instances of NSHS. As shown in the firstexample, sequential hybrid systems can also originate from a switched control strategy, suchas sliding modes, applied to a nonlinear system.

Example 5.1 (Sliding Modes). Consider that a nonlinear system with dynamics described by

x = f (x)+ g(x)u

is given. Suppose we wish to use sliding-mode control [18] to stabilize the trajectories to a givensliding surface s = 0, where s ≡ Cx is a scalar. This is a particular case of a sequential hybridsystem in which the sliding surface divides the state space into two regions. Therefore, in thiscase N = 1 since there is only one model and M = 2 because there will be two controllers,one for each region corresponding to each side of the sliding surface. The task to be performedin both regions is to take the trajectories to the sliding surface and therefore, W1 = W2 = 1,n1 = n2 = 0 so that e1(s) = e2(s) = s. If σ = 1 describes the region where s > 0 and


σ = 2 describes the region where s < 0, then for this particular example the switching functionis hc(x) = hc12(x) = hc21(x) = Cx and the candidate control Lyapunov function is

V = V1 = V2 = 1

2h2

c,

which must verify V > 0, V < −αV γ , α > 0, γ = 12 , according to sliding-mode theory [18].

Therefore, the class of hybrid models described in this paper includes instances of sliding-modecontrol systems as a special case and sliding-mode controller design techniques can be derivedfrom the proposed hybrid sequential controller.

Example 5.2 (Gear Shifting in a Car). Consider the model of a car with two gears defined inExample 2.3. It was shown in that example that if the gear shifting function is designed to dependon the continuous-time state as

σp(t) ={

1 w1(x2(t)) ≥ w2(x2(t)),2 w1(x2(t)) < w2(x2(t)),

then the resulting SBS system is described by hp12(x) = hp21(x) = w1(x2)−w2(x2),

f1 = [x2 0]T , g1 = [0 w1(x2)]T , C1 = C = [0 1], W = 1, n = −vmax

f2 = [x2 0]T , g2 = [0 w2(x2)]T , C2 = C = [0 1].When the car starts, the velocity is zero and the system is in mode σp = 1 because w1(0) >w2(0). Define the error signals e1(x) = w1(x2) − w2(x2) and e2(x) = e(x) = x2 − vmax.The objective of the controller is to switch gears in a sequential way, from gear σp = 1 to gearσp = 2 and then achieve maximum velocity, which is only possible in gear σp = 2. Therefore,this system is a sequential hybrid system when the car starts with a zero initial velocity. Theresults of Theorem 4.1 give

V1 = 1

2e2

1, V2 = 1

2e2

2.

The control input is found using (20), which yields the constraint

u = u(x2) > − α1√2w1(x2)∇12(x2)

> 0, (22)

for gear σp = 1, where ∇12 is defined in (6). Using (20) for gear σp = 2 the control input mustverify

u = u(x2) >α2√

2w2(x2)> 0. (23)

Therefore, according to (21)–(23), the stabilizing control law

u > maxx2

(− α1√

2w1(x2)∇12(x2),

α2√2w2(x2)

)

will achieve x2 = vmax in a finite time treach verifying

treach <

√2

α1+

√2‖x�2 − vmax‖

α2,

where x�2 is defined implicitly by w1(x�2) = w2(x�2). �


Fig. 1. System response for K1 = m1 = m2 = −1.5, K2 = 1.

Example 5.3 (Piecewise-affine Systems as Sequential Systems). Consider the same setting as inExample 3.1 for which N = 2, M = 2. The Lyapunov function is chosen as V1 = 1

2 e21 and

V2 = 12 e2

2 with e1(x), e2(x) defined as in Example 3.1. The control signal will be selected as apiecewise-affine state feedback of the form u = Ki x + mi , i = 1, 2. Imposing the conditions

V2 > 0, V2 < −α2V12

2 and V1 > 0, V1 < −α1V 11 it can be shown that

K2 = 1, m2 = −1 − m2, K1 = m1

where m2√

2 > α2, K1 < −α12 − 1 will asymptotically stabilize the system to the open-loop

unstable equilibrium point x = −1. In fact, using the result (21) of Theorem 4.1, this control law

guarantees that the boundary x = 0 will be achieved in a finite time t <√

2x0α2

whenever the initialcondition x0 is inside regionR2, which is not the region that contains the closed-loop equilibriumpoint (see Fig. 1). When the trajectory hits the boundary, the controller switches to the controlsignal of region R1, which is an asymptotically stabilizing affine controller. Consequently, thesystem converges to the closed-loop equilibrium point using a two-stage control strategy:

• Stage 1. Go from region R2 to the boundary between regions R2 and R1• Stage 2. Go from the boundary to the equilibrium point of region R1.

Example 5.4 (Piecewise-affine Systems as Non-Sequential Systems). Consider the same settingas in Example 3.2 for which N = 2, M = 2. Suppose that the strategy now is to use adynamic output feedback controller designed by searching for a general piecewise-quadraticcontrol Lyapunov function that is continuous at the boundaries with the constraint that thecomponent of the closed-loop vector field perpendicular to the boundary is continuous (to avoidsliding modes at the switching and thus have a provably stabilizing switching). This is theapproach followed in [12]. That work also shows that this control problem can be formulatedas a bi-convex optimization problem and solved numerically for a suboptimal solution. Thecontroller and Lyapunov function parameters, and a plot of the Lyapunov function are foundin [12].


Example 5.5 (Computer-fan Cooling System). Suppose that for the problem described inExample 2.2 an extra dissipator is included in the electronic equipment in addition to the fan.This dissipator is modeled as a heat flow rate u that can be actively changed so that the dynamicsare now described by

x =

⎧⎪⎪⎨⎪⎪⎩

2η


Cthermal(q0 + u) , x ≥ Tref

η


Cthermal(q0 + u) , x < Tref.

(24)

Assume we want to design the control input u such that the temperature of the system convergesexponentially to Ta . Notice that the switching rule is the same as in Example 2.2, i.e., hp12 =hp21 = x − Tref. Consider now the following candidate control Lyapunov function

V1(x) = V2(x) = V (x) = 1

2(x − Ta)

2 = 1

2

(x T Px + 2qT x + r

), (25)

where P = 1, q = −Ta, r = T 2a . V (x) is clearly positive definite. With the choice

u ={−q0 x(t) ≥ Tref,

−q0 − η (x − Ta) x(t) < Tref,

we get V < −αV γ with α = 4η/Cthermal and γ = 1, which proves exponential stabilityof the trajectories to x = Ta . Notice that this closed-loop hybrid system is an NSHS withe1(x) = e2(x) = x − Ta . As required for NSHS, e1(x), e2(x) → 0 when t → ∞. Noticealso that the Lyapunov function (25) is a perfect square, and therefore also a SOS. Therefore,once the controller is designed, the results of Theorem 3.1 can also be invoked to determine thestability of the error trajectories of the closed loop system. �

Example 5.6 (Cart Parking). Consider that the control objective is to use state feedback to parka cart whose kinematic equations in the x–y plane are⎧⎨

⎩x = u cosψ − v sinψy = u sinψ + v cosψψ = r

(26)

at the configuration X ≡ (x, y, ψ) = (0, 0, 0). The input u is the forward velocity of the cartand the input r is the rotational velocity of the cart around the z-axis. The lateral velocity of thecart is v. It will be assumed that the slip coefficient of the terrain and the cart velocities are smallenough. This implies that assuming v = 0 is reasonable. According to [54], the control strategywill be to perform the parking maneuver using three steps:

1. First take the cart to the line that passes through the origin and holds the desired heading,2. Then take the cart along that line to the origin, keeping its heading,3. When the origin is reached make the control signals equal to zero and stop.

This strategy clearly suggests a hybrid model for the closed-loop system. In particular, thediscrete parameter σ can be used to indicate the type of maneuver that is being implemented:

• σ = 1 means the cart is approaching the line that passes through the origin with zero heading(i.e., the line y = 0, ψ = 0)

• σ = 2 means that the cart is going to the origin along that line (i.e., the origin is the place onthe line for which x = 0), and

• σ = 3 means the cart has stopped at the origin (i.e., at x = 0, y = 0, ψ = 0).


In this case, N = 1 and M = 3. Ideally, this would be an SHS whenever the initial condition isoutside the line. However, in general the closed-loop system must be modeled as an NSHS, notonly because the initial condition is arbitrary, but also because disturbances can force this strategyto be repeatedly executed until the convergence of the trajectories to the origin. Therefore, thereis no specific ordering for the tasks but it is desired that as t → ∞, (x, y, ψ) → (0, 0, 0). Forσ = 1, define s = C1 X and for σ = 2, define s = C2 X with X ≡ (x, y, ψ) and

C1 =⎡⎣0 1 0

0 01

λψ

⎤⎦ , C2 = [

1 0 0],

for some constant λψ > 0. Also define W1 = W2 = 1 and n1 = n2 = 0, W3 = n3 = 0, which

imply that e1 = [ yψ

λψ]T , e2 = x and e3 = 0.

The candidate control Lyapunov functions will be chosen as

V1(X) = 1

2eT

1 e1 = 1

2

(y2 + 1

λ2ψ

ψ2

)

and V2(X) = 12 e2

2 = 12 x2. This control strategy and these candidate control Lyapunov functions

are the ones proposed in [54]. The switching strategy described in [54] was based on thesupervisory switching law suggested in [14] with hysteresis switching to avoid chattering andhas the form

h12 (V1(X), V2(X)) = h12(X) = 0, h21 (V1(X), V2(X)) = h21(X) = 0

at a switching from σ = 1 to σ = 2 and σ = 2 to σ = 1, respectively (see [54] for details). Itwas shown in [54] that by making V1 > 0, V1 < 0 for σ = 1 and V2 > 0, V2 < 0 for σ = 2,the closed-loop system is asymptotically stable, even when the lateral velocity v is not zero.The importance of this example is to show that systems with nonholonomic constraints [63] andunderactuated systems [64] with a task-based control law can be modeled as NSHS. Furthermore,using the framework developed in this paper, multiple quadratic Lyapunov functions can be usedto design controllers that are guaranteed to stabilize these systems. Thus, the class of hybridmodels described in this paper also incorporates control systems with nonholonomic constraintsand/or underactuated systems as special cases.

6. Conclusions

The purpose of this paper was to define a new class of hybrid systems – state-based switchedsystems (SBSS) – that can be used to model many engineering applications and to present ageneral unified procedure to perform both analysis and controller synthesis for these systems. Forstatic state and output feedback of a particular subclass of SBSS, called sequential hybrid systems(SHS), the paper proposes the synthesis of a new hybrid sequential sliding-mode controller.It was shown that the unified static state and output feedback control design problem can besolved in a systematic way using multiple Lyapunov functions for SHS. For another subclass,called nonsequential hybrid systems (NSHS), it was shown in the examples that the controldesign problem can still be solved for specific instances using multiple Lyapunov functions.Furthermore, the examples illustrate that many existing design techniques for systems as diverseas sliding modes, piecewise-affine systems, nonholonomic and underactuated systems can beunified under the framework developed in this paper. This presents an important first step toward


a unified analysis and controller synthesis methodology for a large class of important hybridsystems.

7. Acknowledgement

The first author would like to acknowledge the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) for funding his research.

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Toward unified analysis and controller synthesis for a class of hybrid systems

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