- Email: [email protected]
A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control⁎
Available online at www.sciencedirect.com
ScienceDirect IFAC PapersOnLine 52-28 (2019) 44–49
A Comparison of LPV Gain Scheduling and A A Comparison Comparison of of LPV LPV Gain Gain Scheduling Scheduling and and Control Contraction for Nonlinear A Comparison of LPVMetrics Gain Scheduling and Control Contraction Metrics for Nonlinear Control Contraction Metrics for Nonlinear Control Control Contraction Metrics for Nonlinear Control Control Control ∗ Ruigang Wang ∗∗ Roland T´ oth ∗∗ ∗∗ Ian R. Manchester ∗
Ruigang Wang ∗ Roland T´ oth ∗∗ Ian R. Manchester ∗ Ruigang Wang Roland T´ oth Ian R. Manchester ∗ oth ∗∗ Ian R. Manchester ∗ ∗ Ruigang Wang Roland T´ ∗ Australian Centre for Field Robotics, The University of Sydney, Australian Centre for Field Robotics, University of Sydney, ∗ Australian Centre for Field Robotics, The The University of Sydney, NSW 2006, Australia (e-mail: {ruigang.wang, NSW 2006, Australia (e-mail: {ruigang.wang, ∗ Australian Centre forAustralia Field Robotics, University of Sydney, NSW 2006, (e-mail:The {ruigang.wang, ian.manchester}@sydney.edu.au). ∗∗ NSWofian.manchester}@sydney.edu.au). 2006, Australia (e-mail: {ruigang.wang, ian.manchester}@sydney.edu.au). Electrical Engineering, Eindhoven ∗∗ Department Electrical Engineering, Eindhoven University University of of ∗∗ Department of Department ofian.manchester}@sydney.edu.au). Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ∗∗ Department of Box Electrical Engineering, Eindhoven of Technology, P.O. 513, 5600 MB Eindhoven, TheUniversity Netherlands (e-mail: [email protected]). [email protected]). Technology, P.O. Box(e-mail: 513, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]). (e-mail: [email protected]). Abstract: Gain-scheduled Gain-scheduled control control based based on on linear linear parameter-varying parameter-varying (LPV) (LPV) models models derived derived Abstract: Abstract: Gain-scheduled control based on linear parameter-varying (LPV) models derived from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. from local linearizations is a control widespread nonlinear technique for tracking time-varying setpoints. Abstract: Gain-scheduled ononlinear parameter-varying (LPV) models derived from local aalinearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, nonlinear control control schemebased based Control Contraction Metrics (CCMs) has been Recently, nonlinear scheme based on Control Contraction Metrics (CCMs) has been from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, a nonlinear control scheme based on Control Contraction Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of developed tononlinear track arbitrary admissible trajectories. ThisContraction paper presents a comparison study of Recently, aapproaches. control scheme based on Control Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of these two We show that the CCM based approach is an extended gain-scheduled these two approaches. We show that the CCM based approach is an extended gain-scheduled developed to track arbitrary admissible trajectories. This paper presents aperformance comparison study of these approaches. We show that reference-independent the CCM based approach is anand extended gain-scheduled controltwo scheme which achieves global reference-independent stability and through control scheme which achieves global stability performance through these two approaches. We show that the CCM based of approach is an extended gain-scheduled control scheme which achieves global reference-independent stability and performance through an exact control realization which integrates a series local LPV controllers on a particular an exactscheme controlwhich realization which integrates a series of local LPV controllers on a particular control achieves global reference-independent stability and performance through an exact control integrates LPV controllers on a particular path between therealization current andwhich reference states.a series of local path between the current and reference states. an exact control realization which integrates a series of local LPV controllers on a particular path between the current and reference states. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. path between the current and reference states. 1. INTRODUCTION INTRODUCTION controllers 1. controllers on on P P or or parametrizing parametrizing an an LPV LPV controller controller and and 1. INTRODUCTION controllers on P or parametrizing an LPVproblem controller and solving the stabilization and performance jointly solving the stabilization and performance problem jointly 1. applications, INTRODUCTION controllers on P orlocal parametrizing an LPVproblem controller and solving the stabilization and performance jointly over P. Typically, equilibrium-independent stability In many industrial systems with nonlinear P. the Typically, local equilibrium-independent stability In many industrial applications, systems with nonlinear over solving stabilization and performance problem jointly over P. Typically, local equilibrium-independent stability performance can achieved In many industrial applications, with nonlinear dynamical behavior are are required to tosystems be operated operated in aa wide wide and and performance can be be achieved via via these these methods, methods, dynamical behavior required be in over P. Typically, local equilibrium-independent stability In many industrial applications, systems with nonlinear and performance can be achieved via these(Rugh methods, requiring σ to be “sufficiently slow-varying” and dynamical behavior are required to be operated in a wide range of operating conditions. A widespread approach requiring σ to be “sufficiently slow-varying” (Rugh and range of operating conditions. A widespread approach and performance can be achieved via these methods, dynamical behaviorisare required A to be operated in alinear wide Shamma, requiring 2000). σ to be “sufficiently slow-varying” (Rugh and range of operating conditions. widespread approach for this situation gain-scheduled control using Shamma, 2000). for this situation is gain-scheduled control using linear requiring 2000). σ to be “sufficiently slow-varying” (Rugh and range of situation operatingis conditions. Arepresentations widespread approach Shamma, for this gain-scheduled control using linear parameter-varying (LPV) system (Papaparameter-varying (LPV) system representations (PapaIt 2000).to for this situation is gain-scheduled control using linear It is is important important to highlight highlight that that next next to to gain-scheduling gain-scheduling parameter-varying (LPV) system georgiou et al., al., 2000; 2000; Rugh and representations Shamma, 2000;(PapaKlatt Shamma, It is important to highlight that next to gain-scheduling georgiou et Rugh and Shamma, 2000; Klatt based modeling and control, which is often called parameter-varying (LPV) system representations (Papabased modeling and control, which is often called a a local local georgiou et al., 2000; Rugh and Shamma, 2000; Klatt and Engell, Engell, 1998). 1998). The The underlying underlying idea idea is is to to introduce introduce It is important to highlight that next to gain-scheduling based modeling and control, which is often called abased local and LPV approach, modern LPV control methods are georgiou et scheduling al., 2000; Rugh and Shamma, 2000; Klatt LPV approach, modern LPV control methods are based and Engell, 1998). The underlying idea is to introduce a so-called variable σ that indicates the curmodeling and control, is often called local LPV approach, modern LPV control methods areabased a so-called scheduling variable σ that indicates the cur- based on directly directly transforming the which nonlinear system model via and Engell, 1998). The underlying idea is to introduce on transforming the nonlinear system model via arent so-called scheduling σ that theaa current operating point of ofvariable the system system andindicates construct lin- LPV approach, modern LPV control methods are(2010); based on directly transforming the nonlinear system model via operating point the and construct linthe so-called global embedding principle, see T´ o th a so-called scheduling variable σ that indicates the curthe so-called global embedding principle, see T´ o th (2010); rent operating of thethe construct a lin- on ear model model thatpoint describes thesystem local, and linearized dynamics directly transforming the nonlinear system model via the so-called global embedding principle, see T´ o th (2010); ear that describes local, linearized dynamics Hoffmann and Werner (2015) for an overview, and then rent operating point of thethe system and construct a linandglobal Werner (2015) principle, for an overview, and then ear model describes local, linearized dynamics of the the plantthat around each point. The parameters of the Hoffmann the so-called embedding see T´ o th (2010); Hoffmann and Werner (2015) for an overview, and then of plant around each point. The parameters of the synthesizing an LPV law that gives and ear model that describes dynamics anWerner LPV control control law that gives stability stability and of the plant around each point. Thelinearized parameters of σ resulting model varies withthe σ. local, Next, assuming that σthe is synthesizing Hoffmann and (2015) for an overview, and then synthesizing an LPV control law that gives variations stability and resulting model varies with σ. Next, assuming that is performance guarantees over all possible of of the plant around each point. The parameters of the performance guarantees over all possible variations of resulting model varies with σ. Next, assuming that σ is an external variable (independent from the inputs) an synthesizing an LPV control law that gives variations stability and performance guarantees over all possible of an external variable (independent from the inputs) an σ. Such methods had been thought superior over gainresulting model varies with σ. Next, assuming that σ is σ. Such methods had been thought superior over gainan variable (independent from the inputs) an performance LPVexternal controller dependent on σ σ is is designed designed that, by using using guarantees over all possible variations of σ. Such methods had been thought superior over gainLPV controller dependent on that, by scheduling techniques as they provided direct stability an external (independent from the inputs) an scheduling techniques as they provided directover stability LPV controller dependent on and σ is Packard, designed that, by using linear systemvariable theory (Becker and Packard, 1994), ensures σ. Such methods had been thought superior gainscheduling techniques as they provided direct stability linear system theory (Becker 1994), ensures for the nonlinear system following LPV controller dependent on and σ is Packard, designed that, by using guarantees guarantees for the embedded embedded system following linear system theory (Becker 1994), ensures stability and performance performance specifications for the the LPV model scheduling techniques as theynonlinear provided direct stability guarantees for the embedded nonlinear system following stability and specifications for LPV model aa differential inclusion concept. However, recent studies linear system theory (Becker and Packard, 1994), ensures differential inclusion concept. However, recent studies stability and performance specifications for the LPV model under possible variations of σ in a user specified region of guarantees for the embedded nonlinear system following a differential inclusion concept. However, recent studies under possible variations of σ in a user specified region of Scorletti et al. (2015) indicate that performance issues for stability and performance specifications for the LPV model Scorletti et al. (2015) indicate that performance issues for under possible variations of σ in a user specified region of operating conditions conditions P. P. Finally, Finally, aa nonlinear nonlinear control control law law is is aScorletti differential inclusion concept. However, recentissues studies et al. (2015) indicate that performance for operating reference tracking objectives might still be present. To adunder possible variations of with σ in aameasured user specified region of reference tracking objectives might still be present. To adoperating conditions P. Finally, nonlinear control law is obtained by substituting σ information of Scorletti et al. (2015) indicate that still performance issues for reference tracking objectives might be present. To adobtained by substituting σ with measured information of dress this problem, a strong notion incremental stability operating conditions Finally, nonlinearinformation control law of is dress this tracking problem,objectives a strong notion - incremental stability obtained by substituting σsystem. with ameasured the operating operating point of ofP.the the system. reference might still be present. To has address this problem, a strong notion incremental stability the point was considered, and the corresponding LPV modeling obtained by substituting with measured information of was considered, andathe corresponding LPV modeling has the operating point of theσsystem. dress this problem, strong notion - incremental stability was considered, and the corresponding LPV modeling has a connection to the local linearization of the plant. There are many many approaches available the operating pointapproaches of the system. connection to the local linearization of the plant. There are available to to construct construct an an awas considered, the corresponding LPV a connection to and the local linearization of the modeling plant. has There are many approaches available to construct an LPV model of the plant based on this methodology, see LPV model of theapproaches plant basedavailable on this methodology, see builds stability aContraction connectiontheory to the which local linearization of the plant. results There are many to construct an Contraction theory which builds on on global global stability results LPV model of (2013) the plant based on this methodology, see Bachnas et al. al. (2013) for an an overview. Typically, the plant plant Contraction theory which builds on global stability results Bachnas et for overview. Typically, the from local analysis has gained much attention for nonlinear LPV model of the plant based on this methodology, see from local analysis has gained much attention for nonlinear Bachnas et al. (2013) for an overview. Typically, the plant is linearized linearized around around aa given given set set of of equilibrium equilibrium points points Contraction theoryhas which builds on attention global stability results from local analysis gained much for nonlinear is systems (Lohmiller and Slotine, 1998; Forni and Sepulchre, Bachnas et (2013) for an overview. thepoints plant (Lohmiller andgained Slotine, 1998; Forni and Sepulchre, is linearized a given set set of Typically, equilibrium (griding of al. P)around and the the resulting set of LTI models models are systems from local analysis has much attention for nonlinear systems (Lohmiller and Slotine, 1998; Forni and Sepulchre, (griding of P) and resulting of LTI are 2014). Related works include velocity linearization (Leith is linearized around given set set ofis of equilibrium points Related works velocity linearization (Leith (griding and resulting LTI models are 2014). of P) interpolated over P the ora linearization linearization accomplished over systems (Lohmiller andinclude Slotine, 1998; Forni and Sepulchre, 2014). Related works include linearization (Leith interpolated over P or is of accomplished over and Leithead, 2000) and Gˆ aavelocity teaux derivative (Fromion (griding of P) and the resulting set LTI models are and Leithead, 2000) and Gˆ teaux derivative (Fromion interpolated over P or linearization is accomplished over input and state trajectories. Similarly, the LPV controller 2014). Related works include velocity linearization (Leith and Leithead, 2000) and Gˆ a teaux derivative (Fromion input and state trajectories. Similarly, the LPV controller et al., 2001; Fromion and Scorletti, 2003). Recently, coninterpolated overtrajectories. Pdesigning or linearization is the accomplished over et al.,Leithead, 2001; Fromion and Scorletti, 2003). Recently, coninput and state Similarly, LPV controller can be obtained by LTI controllers separately for and 2000) and Gˆ a teaux derivative (Fromion et al., 2001; Fromion and Scorletti, 2003). Recently, concan be obtained by designing LTI controllers separately for traction analysis was extended to constructive nonlinear input and state trajectories. Similarly, the LPV controller traction analysis was extended to constructive nonlinear can be obtained by designing LTI controllers separately for finite set of values of σ and then interpolating these LTI et al., 2001; Fromion and 2003). Recently, contraction analysis extended to constructive nonlinear finitebeset of values of σ and then interpolating these LTI control design bywas using aScorletti, differential version of control can obtained by designing LTI controllers separately for control design by using a differential version of control finite set of values of σ and then interpolating these LTI traction analysis was extended to constructive nonlinear control design by using a differential version of control Lyapunov function Control Contraction Metric (CCM) work supported by the Australian Research Council. finite of was values of σ and then interpolating these LTI Lyapunov function - Control Contraction Metric This This set work was supported by the Australian Research Council. control design bySlotine, using a2017, differential version of (CCM) control Lyapunov function - Control Contraction Metric (CCM) (Manchester and 2018). Further extensions This work has received funding European Research Council This work was supported byfrom thethe Australian Research Council. (Manchester and Slotine, 2017, 2018). Further extensions This work has received funding from the European Research Council Lyapunov function Control Contraction Metric (CCM) (Manchester and Slotine, 2017,include 2018). distributed Further extensions of the approach control This work thethe Australian Research Council. (ERC) under the supported European Unions Horizon 2020 research and This work haswas received fundingby from European Research Council of the CCM CCM based based approach control (ERC) under the European Unions Horizon 2020 research and (Manchester Slotine, 2017,include 2018). distributed Further extensions of the CCM and based approach include distributed control This work has received funding from theHorizon European Research Council innovation programme (grant agreement nr. 714663). (ERC) under the European Unions 2020 research and innovation programme (grant agreement nr. 714663). of the CCM based approach include distributed control (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement nr. 714663). innovation programme agreement nr. 714663). 2405-8963 © 2019, IFAC(grant (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.346
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
(Shiromoto et al., 2018) and distributed economic model predictive control (MPC) (Wang et al., 2017). The main contribution of this paper is a comparison study between the CCM based nonlinear control approach and the LPV gain scheduling technique using local linearization. For simplicity, only state-feedback control design is considered. We show that CCM based control is an extended LPV gain scheduling approach. First, the so-called differential dynamics in contraction theory can be seen as a local LPV model which takes linearization along any admissible solution rather than an equilibrium family in conventional gain-scheduling. Second, similar parameterdependent linear matrix inequality (LMI) conditions are derived as in local LPV synthesis. One difference is that the CCM based approach explicitly takes the original nonlinear plant into account leading to less conservative results. Furthermore, the control realization integrates a series of local controllers on a particular path joining the current and reference state trajectory, which leads to an exact realization without any hidden coupling term. Based on this, local stability and performance design can be carried onto the entire state space as the length of the path shrinks exponentially. Paper outline. Section 2 presents formulations of different stability and performance. Section 3 gives a brief review of the LPV gain scheduling approach using local linearization, which is mainly adopted from Rugh and Shamma (2000). Section 4 discusses the various connections and extensions between CCM and LPV based approaches. An illustrative example is presented in Section 5. 2. PRELIMINARIES Let |x| be the Euclidean norm of a vector x. For any matrix A ∈ Rn×n , we use the notation He{A} := A + A . Positive (negative) definiteness of a Hermitian matrix X is denoted as X 0 (X ≺ 0). C k denotes the set of vector signals on R which are k th times differentiable. L2 is the space of square-integrable vector signals on R≥0 , i.e., ∞ |f (t)|2 dt < ∞. The causal truncation (·)T is f := 0 defined by (f )T (t) := f (t) for t ≤ T and 0 otherwise. Le2 is the space of vector signals on R≥0 whose causal truncation belongs to L2 .
In this paper, we consider a nonlinear system x˙ = f (x, u, w), z = h(x, u, w) (1) nx nu nw where x(t) ∈ R , u(t) ∈ R , w(t) ∈ R , z(t) ∈ Rnz are state, control, external input and performance output signals at time t ∈ R≥0 , respectively. The functions f and h are assumed to be smooth and time-invariant. We define a target trajectory to be a forward-complete solution of (1), i.e., a pair (x∗ , u∗ , w∗ , z ∗ )(·) with x∗ (·) piecewise differentiable and (u∗ , w∗ , z ∗ )(·) piecewise continuous satisfying (1) for all t ∈ R≥0 . The target trajectory is said to be an equilibrium if (x∗ , u∗ , w∗ , z ∗ )(·) = (xe , ue , we , ze ). For simplicity, we assume that the nominal external input is w∗ (·) = 0. We will consider state-feedback controllers of the form u(t) = κ(x(t), x∗ (t), u∗ (t)). (2) To define a nonlinear control problem precisely, we must be specific about the notion of stability and performance.
45
The closed-loop system of (1) and (2) is said to be globally asymptotically stable with respect to the target trajectory (x∗ , u∗ )(·) if the closed-loop solution x(t) of the nominal system (i.e., w = w∗ ) exists and satisfies 1) For any there exists an ρ such that |x(0)−x∗ (0)| < ρ implies |x(t) − x∗ (t)| < , 2) For any initial condition x(0) ∈ Rn , the closed-loop solution satisfies |x(t) − x∗ (t)| → 0.
Global exponential stability is a stronger notion and requires that there exists a R and λ such that |x(t) − x∗ (t)| ≤ Re−λt |x(0) − x∗ (0)|, ∀x(0) ∈ Rn . (3) The closed-loop system is said to achieve L2 -gain performance of α for the target trajectory (x∗ , u∗ , w∗ , z ∗ )(·), if for any initial condition x(0) and input w such that w − w∗ ∈ Le2 , and for all T > 0 solutions exist and satisfy (4) (z − z ∗ )T 2 ≤ α2 (w − w∗ )T + β(x(0), x∗ (0)) where β(x1 , x2 ) ≥ 0 with β(x, x) = 0.
The above reference-related stability and performance notions allow us to investigate different formulations of control objectives in a unified way. Let B∗ be the set of target trajectories which are used for the definitions of asymptotic (exponential) stability and L2 performance. In a standard formulation, B∗ only contains one “preferred” trajectory – the zero solution (i.e., f (0, 0, 0) = 0, h(0, 0, 0) = 0). A stronger formulation, called equilibrium-independent asymptotic (exponential) stability and L2 gain, is referred to the case where B∗ is chosen to be the set of all possible equilibrium points (Simpson-Porco, 2019). The socalled universal exponential stability and L2 gain is an even stronger formulation which requires B∗ to include all admissible trajectories of the nominal system (Manchester and Slotine (2017, 2018)). The incremental formulation is referred to the case where asymptotic (exponential) stability and L2 gain are satisfied for any pair of system trajectories. Note that universal exponential stability is equivalent to incremental exponential stability. But universal L2 gain is weaker than the incremental one (Manchester and Slotine, 2018). 3. GAIN SCHEDULING APPROACH 3.1 System Linearization To construct an LPV representation for (1) using local modelling concept, we assume that the equilibrium points are uniquely characterized by xe . Hence, to describe the equilibrium points associated local dynamics of the system, we introduce a scheduling variable σ ∈ Rnσ that depends on the state, i.e., σ = g(x) (5) where g is a smooth vector function. Note that σ can also depend on w if it is measurable. Here the possible trajectories of the scheduling signal σ(t) are assumed to belong to the set ˙ ∀t ≥ 0} ˙ ∈ P, (6) T = {σ ∈ C 1 : σ(t) ∈ P, σ(t) nσ nσ ˙ where P = {σ ∈ R : |σi | ≤ σ i } and P = {p ∈ R : |pi | ≤ pi } with 1 ≤ i ≤ nσ . Using σ, the equilibrium family is characterized as the set {(xe , ue , ze , we )(σ)}σ∈P where xe (·), ue (·), ze (·), we (·) are smooth functions, and (xe , ue , ze , we )(σ) is an equilibrium of (1) for all σ ∈ P.
46
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
By linearizing (1) around the equilibrium family, we obtain an LPV model as follows: xδ A(σ) Bu (σ) Bw (σ) x˙ δ uδ , σ ∈ P (7) = C(σ) Du (σ) Dw (σ) zδ wδ where xδ = x − xe (σ), uδ = u − ue (σ), wδ = w − we (σ), zδ = z − ze (σ) are deviation variables. The matrices A, Bu , Bw , C, Du , Dw are defined as the evaluations of ∂f ∂f ∂f ∂h ∂h ∂h ∂x , ∂u , ∂w , ∂x , ∂u , ∂w at the σ defined equilibrium point.
σ( x e(
t))
xe (σ)
Fig. 1. Illustration of gain-scheduling control.
3.2 Control Synthesis Consider the static LPV controller of the form uδ = K(σ)xδ , which yields a closed-loop LPV system x˙ δ A(σ) B(σ) xδ = C(σ) D(σ) wδ zδ
x(t)
(8)
(9)
with A = A + Bu K, B = Bw , C = C + Du K, D = Dw . Theorem 1. The unforced closed-loop system (10) x˙ δ = A(σ)xδ , σ ∈ T is exponentially stable if there exists a M (σ) 0 such that nσ ∂M (σ) He{M (σ)A(σ)} + 2λM (σ) + ρi ≺ 0 (11) ∂σi i=1 ˙ for all σ ∈ P and ρ ∈ P.
The above theorem implies that V (xδ , σ) := x δ M (σ)xδ is a parameter-dependent Lyapunov function for system (10). By applying a congruence transformation (12) W (σ) = M −1 (σ), L(σ) = K(σ)W (σ), we can obtain a convex formulation: nσ ∂W (σ) He{A(σ)W (σ)+Bu (σ)L(σ)}+2λW (σ)− ρi ≺0 ∂σi i=1 (13) ˙ for all σ ∈ P and ρ ∈ P.
Note that due to linearity of the LPV description (9), exponential stability and L2 gain bound of (9) w.r.t. the origin is equivalent to the equilibrium-independent asymptotic stability and L2 gain (Rugh and Shamma, 2000; Hoffmann and Werner, 2015). Theorem 2. A controller (8) achieves an L2 performance level of α for LPV system (7) if there exists M (σ) 0 ˙ such that, for all σ ∈ P and ρ ∈ P, −1 M(σ, ρ) M (σ)B(σ) α C (σ) B (σ)M (σ) (14) −I α−1 D (σ) ≺ 0 −1 −1 α C(σ) α D(σ) −I nσ (σ) ρi ∂M where M(σ, ρ) = He{M (σ)A(σ)} + i=1 ∂σi . Based on the above condition, we can synthesize an LPV controller which achieves the minimal L2 -gain bound for the closed-loop LPV system. 3.3 Controller Realization The LPV control realization problem is to construct a gain-scheduled law u = κ(x, σ) such that
ue (σ) = κ(xe (σ), σ), (15a) ∂κ (xe (σ), σ) = K(σ). (15b) ∂x Condition (15b) implies that linearization of u = κ(x, σ) at this equilibrium is the LPV controller (8). An intuitive choice of control realization in the literature is u = ue (σ) + K(σ)[x − xe (σ)]. (16) Under the assumption that the equilibrium points of (1) are uniquely characterized by xe , σ can be expressed in terms of x via (5). Using this relation, (16) reads as u = ue (g(x)) + K(g(x))[x − xe (g(x))]. (17) The main “trick” behind of this gain-scheduling approach is that σ is treated as a parametric/dynamic uncertainty throughout the design process, but during controller realization is substituted by a function of a measured variable characterizing the operating point changes (Rugh and Shamma (2000)). Although σ is implicitly involved via equilibrium parameterizations, linearization of (17) may not satisfy condition (15b) since ∂xe (σ) ∂g ∂ue (σ) uδ = K(σ)xδ + − K(σ) (xe (σ))xδ ∂σ ∂σ ∂x contains additional terms, called hidden coupling terms, compared with (8). These terms may lead to closed-loop instability regardless the fact that exponential stability is achieved in the control synthesis stage, which is a well-known drawback of the local LPV controller (see Example 8 in Rugh and Shamma (2000)). 3.4 Stability and Performance Assessment The core idea of gain-scheduled control (17) is to track a reference xe (σ(t)) lying on the equilibrium manifold, as shown in Fig. 1. This strategy achieves local equilibriumindependent stability if the schedule signal σ(t) is sufficiently “slowly varying” (Rugh and Shamma, 2000). The main reason is that the scheduled reference trajectory (xe , ue , we , ze )(σ(t)) is not admissible to the closed-loop system x˙ = f (x, κ(x, σ), w) with σ = g(x) since simple substitution yields a residual term E(σ)σ(t) ˙ with E(σ) = ∂xe (σ) . Therefore, the actual linearization of the closed∂σ loop system with wδ (t) = 0 is x˙ δ = A(σ)xδ − E(σ)σ. ˙ (18) If the rates of parameter variation are not “sufficiently slow”, the residual terms can drive the state away from the small neighborhood of xe (σ), which may violate the local stability design. To ensure global stability and performance, excessive simulations or even experiments are needed.
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
4. CCM BASED NONLINEAR CONTROL In this section, we will show the connections and differences between the CCM-based control and local LPVbased gain-scheduled control. Both approaches use very close LPV modeling and control synthesis formulations. The major differences come from the ways to interpret and use the LPV tools. In the CCM-based approach, LPV modeling, design and realization are carried out for the entire state space, which is an extension to the local gainscheduling approach as it only considers the equilibrium manifold (as shown in Fig. 1). With this conceptual innovation, the CCM-based approach can provide an exact control realization which does not contain any hidden coupling term and achieve universal stability and L2 performance.
Firstly, we recall some basic facts of Riemannian geometry from Do Carmo (1992). A Riemannian metric on Rn is a smooth matrix function M (x) 0 which defines an inner product δ1 , δ2 x = δ1 M (x)δ2 for any two tangent vector δ1 , δ2 . A metric is called uniformly bounded if there exist positive constants a2 ≥ a1 such that a1 I ≺ M (x) ≺ a2 I, ∀x ∈ Rn . Γ(x0 , x1 ) denotes the set of piecewise smooth paths c : [0, 1] → Rn with c(0) = x0 and c(1) = x1 . The curved length and energy of c(·) is defined by 1 1 cs , cs c(s) ds and ε(c) = cs , cs c(s) ds (c) = 0
0
where cs = ∂c/∂s, respectively. The geodesic γ(·) denotes a path with the minimal length, i.e., (γ) = inf c∈Γ(x0 ,x1 ) (c). The Riemann distance and energy between x0 and x1 are defined by d(x0 , x1 ) = (γ) and ε(x0 , x1 ) = ε(γ) = 2 (γ).
4.1 System Linearization By choosing σ = (x, u, w), we can construct a continuously linearized system (so-called differential dynamics) δ δ˙x = A(σ) Bu (σ) Bw (σ) δx (19) u C(σ) Du (σ) Dw (σ) δz δw where the matrices A, Bu , Bw , C, Du , Dw are defined in a similar way as in the local gain-scheduling approach. The variables δx , δu , δw , δz are the virtual displacement between neighboring solutions (Lohmiller and Slotine, 1998) or the tangent vector of the solution manifold (Forni and Sepulchre, 2014). Other related linearization techniques include velocity linearization (Leith and Leithead, 2000) and Gˆ ateaux derivative (Fromion et al., 2001). Remark 3. Note that the differential dynamics (19) can be seen as a local LPV system defined on the entire state space rather than the equilibrium manifold. 4.2 Control Synthesis The control synthesis searches for a differential controller: (20) δu = K(σ)δx := K(x, u)δx which stabilize the unforced closed-loop dynamics (21) δ˙x = A(σ)δx := [A(σ) + Bu (σ)K(σ)]δx .
It can be achieved by a sufficient condition as follows nx ∂M (x) fi (σ) ≺ 0 (22) He{M (x)A(σ)} + 2λM (x) + ∂xi i=1
47
where fi is the model of the state xi . The above synthesis formulation is very close to (11). Thus, similar convexation technique can be applied here. The main difference between these two formulation is that (22) uses detailed model (1) to describe the scheduling signal σ(t) while (13) uses coarse description (6) which only contains the region of the parameter and its variation. This can lead to less conservative results. For instance, a non-uniform metric M (x) can be found even if both the parameter σ and its variation σ˙ are unbounded, e.g., the system x˙ = −x − x3 admits a non-uniform metric M (x) = 1 + 3x2 . Here V (x, δx ) = δx M (x)δx is called control contraction metric (CCM), which is a differential control Lyapunov function validated everywhere in the state space. Since the schedule variable σ in the gain scheduling approach only depends on x, the control Lyapunov matrix M (σ) obtained from (11) can be seen as a CCM defined on the equilibrium manifold. For performance analysis, we can obtain a formulation similar to (14): M M B α−1 C B M −I α−1 D ≺ 0 (23) α−1 C α−1 D −I nx (x) fi (σ) ∂M where M = He{M (x)A(σ)} + i=1 ∂xi . 4.3 Controller Realization
As discussed in Section 3.3, the differential controller (20) is generally not completely integrable, i.e., there does not exist a gain-scheduled law κ(·) whose Jacobian is K. Unlike the LPV control, the CCM based approach only considers a much weaker condition - the path-integrability of K. Let γ be a geodesic (a minimal path) joining x∗ (t) and x(t), which can be obtained by solving a simple model predictive control problem online (Leung and Manchester, 2017). The state-feedback law is the integral of the differential control (20) over the path γ: (24) u(t) = κ(x(t), x∗ (t)) := κγ (t, 1) where κγ (t, s) is the unique solution of the following integral equation s ∗ κγ (t, s) := u (t) + K(γ(s), κγ (t, s))γs (s)ds. (25) 0
Remark 4. Note that κγ (t, s) satisfies κγ (t, 0) = u∗ and ∂κγ (t, s) = K(γ(s), κγ (t, s))γs (s), ∀s ∈ [0, 1]. ∂s Thus, it does not contain any hidden coupling term and serves as an exact realization for the differential controller (20). Moreover, it can be also applied to those approaches using incremental analysis (Fromion and Scorletti, 2003) where exact realization for general nonlinear systems is an open problem (Scorletti et al., 2015).
We give a geometric interpretation about the interconnection between the path of control signal κγ (t, s) and the local LPV controller (16). Let 0 = s0 < s1 < · · · < sN = 1 with sj+1 − sj be sufficiently small. For any frozen time t, the integral equation (25) gives κγ (sj+1 ) ≈ κγ (sj ) + K(γ(sj ), κγ (sj ), w)[γ(sj+1 ) − γ(sj )],
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
48
x(
γs sj+1
t)
γ(s)
sj x∗ (t)
Fig. 2. Geometric illustration of the control realization in CCM based approach. where the argument t is omitted for simplicity. Thus, κγ (sj+1 ) is an LPV controller (16) that stabilizes the state γ(sj+1 ) around γ(sj ), as shown in Fig. 2. Based on this observation, κγ integrates a series of local LPV controllers (20) along a particular path γ and the CCM based gain scheduling law (24) is the corresponding control action to the measured state x(t) = γ(t, 1). 4.4 Stability and Performance Assessment Since the control realization (25) is exact, the differential dynamics of the closed-loop system along the path γ is same as (21), which is exponentially stable: d (δ M (x)δx ) = δx M˙ δx + 2δx M Aδx ≤ −2λδx M (x)δx . dt x The global stability follows by integrating the above inequality along a geodesic γ: d ε(x∗ (t), x(t)) ≤ −2λε(x∗ (t), x(t)). (26) dt An explanation using LPV concepts is given as follows. The smooth path γ can be understood as a “chain” of many states joining the current state x(t) to the reference point x∗ (t), and the tangent vector δx = γs as a “link” whose behavior is described by the closed-loop differential dynamics (21), as shown in Fig. 2. The convergence of x(t) to x∗ (t) can be inferred since each link in the chain gets shorter due to local stability. The following theorems give the global stability and performance results for the CCM based approach. Theorem 5. (Manchester and Slotine (2017)). If there exists a uniformly bounded metric a1 I ≤ M (x) ≤ a2 I for which (22) holds for all x, u, w, then system (1) under the controller (24) is universal exponentially stable with rate a2 λ and overshoot C = a1 . Theorem 6. (Manchester and Slotine (2018)). The closedloop system of (1) and (24) has an universally L2 -gain bound α if there exists a uniformly bounded metric M (x) such that (23) holds for all x, u, w. 5. CASE STUDY Consider the following nonlinear system −x1 − x2 + w x˙ 1 = (27) x˙ 2 1 − e−x2 + u where w is a measurable reference signal. This example was used in Rugh (1991) to illustrate gain scheduling design for nonlinear systems. Assume that the control task is to stabilize the system at the equilibrium family 0 xe = , ue = e−we − 1. (28) we
In this section, we compare tracking control of (27) with time-varying we using LPV and CCM-based approaches. For LPV design, we introduce the scheduling as σ = e−we which in terms of the equilibrium point relation is equivalent with σ = e−x2,e . We can obtain an LPV model of the system with coefficient matrices −1 −1 0 A(σ) = , B(σ) = . (29) 0 σ 1
To derive a gain-scheduling controller, we consider placing both the closed-loop eigenvalues at −2, leading to K(σ) = [1 −3 − σ] . (30) Then, the control law (16) with the choice of σ = e−w corresponds to the nonlinear controller u = ue (w) + x1 − (3 + e−w )(x2 − w) (31) which is referred as Gain-Scheduled Controller (GSC) 1. The differential dynamics of the closed-loop system can be represented by −1 −1 δ˙x = A(x)δx = (32) δ 1 a(x2 , w) x where a(x2 , w) = e−x2 − e−w − 3. Since A(x) has positive eigenvalues if x2 < − ln(4 + e−w ), the closed-loop system is unstable in this region.
By implementing the scheduling law according to the equilibrium relation σ = e−x2 , (17) gives the controller GSC 2 as follows u = x1 + e−x2 − 1. (33) Linearization of the closed-loop system at the reference xe (w(t)) = [0, w(t)] yields −1 −1 0 x˙ δ = xδ − . (34) 1 0 w˙
The closed-loop system is globally exponential stable. But the √ differential dynamics have eigenvalues λ1,2 = −1/2 ± 3/2i with larger real parts than the specified ones λ1,2 = −2. This mismatch is caused by the hidden coupling terms: ∂xe (σ) ∂κ(xe (σ), σ) ∆ ∂ue (σ) Kh (σ) = −K(σ) − = 3. (35) ∂σ ∂σ ∂σ For CCM based control design, we choose the following differential state feedback control δu = K(x)δx (36) with K(x) = 1 −(3 + e−x2 ) . This leads to exponentially stable closed-loop differential dynamics with the same eigenvalues as the LPV controller. Thus, we can obtain a constant CCM for the closed-loop system, which implies that the geodesic between x∗ and x is a straight line (i.e., γ(s) = (1 − s)x∗ + sx). Further, the CCM controller can be computed as 1 K(γ(s))(x − x∗ )ds u = u∗ + (37) 0 = x1 + e−x2 − 1 − 3(x2 − w) + w˙ where the target trajectory is x∗ (t) = [0, w(t)] , u∗ (t) = ˙ The closed-loop system e−w(t) − 1 + w(t). d x1 −1 −1 x1 = (38) 1 −3 x2 − w dt x2 − w is globally exponential stable at x∗ (t).
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
Piecewise constant reference
x2
5
0 Ref GSC 1 GSC 2 CCM
-5
0
5
10
15
20
25
Time-varying reference
x2
5
0 Ref GSC 1 GSC 2 CCM
-5
0
5
10
15
time
Fig. 3. Closed-loop trajectories for piecewise-constant and time-varying reference. For comparison study, we consider two tracking scenarios: piecewise-constant setpoints and time-varying references. As shown in Fig. 3, the closed-loop system under GSC 1 is not stable when the state x2 enters into certain regions. Although the controller GSC 2 can ensure equilibriumindependent stability, the transitions are much slower and contain oscillations, compared with other controllers. This is caused by the hidden coupling terms in (35). For timevarying references, GSC 2 cannot reach zero error due to the term w˙ in (34). The CCM based approach overcomes these issues and achieves universal stability. 6. CONCLUSION In this paper, we investigated the apparent connection between contraction theory based nonlinear controller design and the gain-scheduling approach which corresponds to LPV control based on local linearization of the nonlinear system. We show that the CCM based control is an extended LPV gain scheduling approach as it yields a control realization without any hidden coupling term and achieves universal stability. REFERENCES Bachnas, A.A., T´ oth, R., Mesbah, A., and Ludlage, J. (2013). A review on data-driven linear parametervarying modeling approaches: A high-purity distillation column case study. J. Process Control, 24(4), 272–285. Becker, G. and Packard, A. (1994). Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Syst. Control Lett., 23(3), 205–215. Do Carmo, M.P. (1992). Riemannian Geometry. Springer, Bosten, USA.
49
Forni, F. and Sepulchre, R. (2014). A differential lyapunov framework for contraction analysis. IEEE Trans. Autom. Control, 3(59), 614–628. Fromion, V., Monaco, S., and Normand-Cyrot, D. (2001). The weighted incremental norm approach: from linear to nonlinear H∞ control. Automatica, 37(10), 1585–1592. Fromion, V. and Scorletti, G. (2003). A theoretical framework for gain scheduling. Int. J. Robust Nonlin. Control, 13(10), 951–982. Hoffmann, C. and Werner, H. (2015). A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations. IEEE Trans. on Control Syst. Techno., 23(2), 416–433. Klatt, K.U. and Engell, S. (1998). Gain-scheduling trajectory control of a continuous stirred tank reactor. Comput. & Chem. Eng., 22(4), 491–502. Leith, D.J. and Leithead, W.E. (2000). Survey of gainscheduling analysis and design. Int. J. Control, 73(11), 1001–1025. Leung, K. and Manchester, I.R. (2017). Nonlinear stabilization via control contraction metrics: A pseudospectral approach for computing geodesics. In Proc. Amer. Control Conf. (ACC), 1284–1289. Seattle, WA. Lohmiller, W. and Slotine, J.J.E. (1998). On contraction analysis for non-linear systems. Automatica, 34(6), 683– 696. Manchester, I.R. and Slotine, J.J.E. (2017). Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design. IEEE Trans. Autom. Control, 62(6), 3046–3053. Manchester, I.R. and Slotine, J.J.E. (2018). Robust control contraction metrics: A convex approach to nonlinear state-feedback H∞ control. IEEE Control Syst. Lett., 2(3), 333–338. Papageorgiou, G., Glover, K., D’Mello, G., and Patel, Y. (2000). Taking robust LPV control into flight on the VAAC Harrier. In Proc. Conf. on Decision and Control (CDC), 4558–4564. Sydney, Australia. Rugh, W.J. (1991). Analytical framework for gain scheduling. IEEE Control Syst. Mag., 11, 74–84. Rugh, W.J. and Shamma, J.S. (2000). Research on gain scheduling. Automatica, 36, 1401–1425. Scorletti, G., Fromion, V., and De Hillerin, S. (2015). Toward nonlinear tracking and rejection using LPV control. In Proc. IFAC Workshop on Linear Parameter Varying Systems, 13–18. Grenoble, France. Shiromoto, H.S., Revay, M., and Manchester, I.R. (2018). Distributed nonlinear control design using separable control contraction metrics. IEEE Trans. Control Netw. Syst. doi:10.1109/TCNS.2018.2885270. Simpson-Porco, J.W. (2019). Equilibrium-independent dissipativity with quadratic supply rates. IEEE Trans. Autom. Control, 64(4), 1440–1455. T´oth, R. (2010). Modeling and Identification of Linear Parameter-Varying Systems. Springer, Heidelberg. Wang, R., Manchester, I.R., and Bao, J. (2017). Distributed economic MPC with separable control contraction metrics. IEEE Control Syst. Lett., 1(1), 104–109.
ScienceDirect IFAC PapersOnLine 52-28 (2019) 44–49
A Comparison of LPV Gain Scheduling and A A Comparison Comparison of of LPV LPV Gain Gain Scheduling Scheduling and and Control Contraction for Nonlinear A Comparison of LPVMetrics Gain Scheduling and Control Contraction Metrics for Nonlinear Control Contraction Metrics for Nonlinear Control Control Contraction Metrics for Nonlinear Control Control Control ∗ Ruigang Wang ∗∗ Roland T´ oth ∗∗ ∗∗ Ian R. Manchester ∗
Ruigang Wang ∗ Roland T´ oth ∗∗ Ian R. Manchester ∗ Ruigang Wang Roland T´ oth Ian R. Manchester ∗ oth ∗∗ Ian R. Manchester ∗ ∗ Ruigang Wang Roland T´ ∗ Australian Centre for Field Robotics, The University of Sydney, Australian Centre for Field Robotics, University of Sydney, ∗ Australian Centre for Field Robotics, The The University of Sydney, NSW 2006, Australia (e-mail: {ruigang.wang, NSW 2006, Australia (e-mail: {ruigang.wang, ∗ Australian Centre forAustralia Field Robotics, University of Sydney, NSW 2006, (e-mail:The {ruigang.wang, ian.manchester}@sydney.edu.au). ∗∗ NSWofian.manchester}@sydney.edu.au). 2006, Australia (e-mail: {ruigang.wang, ian.manchester}@sydney.edu.au). Electrical Engineering, Eindhoven ∗∗ Department Electrical Engineering, Eindhoven University University of of ∗∗ Department of Department ofian.manchester}@sydney.edu.au). Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ∗∗ Department of Box Electrical Engineering, Eindhoven of Technology, P.O. 513, 5600 MB Eindhoven, TheUniversity Netherlands (e-mail: [email protected]). [email protected]). Technology, P.O. Box(e-mail: 513, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]). (e-mail: [email protected]). Abstract: Gain-scheduled Gain-scheduled control control based based on on linear linear parameter-varying parameter-varying (LPV) (LPV) models models derived derived Abstract: Abstract: Gain-scheduled control based on linear parameter-varying (LPV) models derived from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. from local linearizations is a control widespread nonlinear technique for tracking time-varying setpoints. Abstract: Gain-scheduled ononlinear parameter-varying (LPV) models derived from local aalinearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, nonlinear control control schemebased based Control Contraction Metrics (CCMs) has been Recently, nonlinear scheme based on Control Contraction Metrics (CCMs) has been from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, a nonlinear control scheme based on Control Contraction Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of developed tononlinear track arbitrary admissible trajectories. ThisContraction paper presents a comparison study of Recently, aapproaches. control scheme based on Control Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of these two We show that the CCM based approach is an extended gain-scheduled these two approaches. We show that the CCM based approach is an extended gain-scheduled developed to track arbitrary admissible trajectories. This paper presents aperformance comparison study of these approaches. We show that reference-independent the CCM based approach is anand extended gain-scheduled controltwo scheme which achieves global reference-independent stability and through control scheme which achieves global stability performance through these two approaches. We show that the CCM based of approach is an extended gain-scheduled control scheme which achieves global reference-independent stability and performance through an exact control realization which integrates a series local LPV controllers on a particular an exactscheme controlwhich realization which integrates a series of local LPV controllers on a particular control achieves global reference-independent stability and performance through an exact control integrates LPV controllers on a particular path between therealization current andwhich reference states.a series of local path between the current and reference states. an exact control realization which integrates a series of local LPV controllers on a particular path between the current and reference states. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. path between the current and reference states. 1. INTRODUCTION INTRODUCTION controllers 1. controllers on on P P or or parametrizing parametrizing an an LPV LPV controller controller and and 1. INTRODUCTION controllers on P or parametrizing an LPVproblem controller and solving the stabilization and performance jointly solving the stabilization and performance problem jointly 1. applications, INTRODUCTION controllers on P orlocal parametrizing an LPVproblem controller and solving the stabilization and performance jointly over P. Typically, equilibrium-independent stability In many industrial systems with nonlinear P. the Typically, local equilibrium-independent stability In many industrial applications, systems with nonlinear over solving stabilization and performance problem jointly over P. Typically, local equilibrium-independent stability performance can achieved In many industrial applications, with nonlinear dynamical behavior are are required to tosystems be operated operated in aa wide wide and and performance can be be achieved via via these these methods, methods, dynamical behavior required be in over P. Typically, local equilibrium-independent stability In many industrial applications, systems with nonlinear and performance can be achieved via these(Rugh methods, requiring σ to be “sufficiently slow-varying” and dynamical behavior are required to be operated in a wide range of operating conditions. A widespread approach requiring σ to be “sufficiently slow-varying” (Rugh and range of operating conditions. A widespread approach and performance can be achieved via these methods, dynamical behaviorisare required A to be operated in alinear wide Shamma, requiring 2000). σ to be “sufficiently slow-varying” (Rugh and range of operating conditions. widespread approach for this situation gain-scheduled control using Shamma, 2000). for this situation is gain-scheduled control using linear requiring 2000). σ to be “sufficiently slow-varying” (Rugh and range of situation operatingis conditions. Arepresentations widespread approach Shamma, for this gain-scheduled control using linear parameter-varying (LPV) system (Papaparameter-varying (LPV) system representations (PapaIt 2000).to for this situation is gain-scheduled control using linear It is is important important to highlight highlight that that next next to to gain-scheduling gain-scheduling parameter-varying (LPV) system georgiou et al., al., 2000; 2000; Rugh and representations Shamma, 2000;(PapaKlatt Shamma, It is important to highlight that next to gain-scheduling georgiou et Rugh and Shamma, 2000; Klatt based modeling and control, which is often called parameter-varying (LPV) system representations (Papabased modeling and control, which is often called a a local local georgiou et al., 2000; Rugh and Shamma, 2000; Klatt and Engell, Engell, 1998). 1998). The The underlying underlying idea idea is is to to introduce introduce It is important to highlight that next to gain-scheduling based modeling and control, which is often called abased local and LPV approach, modern LPV control methods are georgiou et scheduling al., 2000; Rugh and Shamma, 2000; Klatt LPV approach, modern LPV control methods are based and Engell, 1998). The underlying idea is to introduce a so-called variable σ that indicates the curmodeling and control, is often called local LPV approach, modern LPV control methods areabased a so-called scheduling variable σ that indicates the cur- based on directly directly transforming the which nonlinear system model via and Engell, 1998). The underlying idea is to introduce on transforming the nonlinear system model via arent so-called scheduling σ that theaa current operating point of ofvariable the system system andindicates construct lin- LPV approach, modern LPV control methods are(2010); based on directly transforming the nonlinear system model via operating point the and construct linthe so-called global embedding principle, see T´ o th a so-called scheduling variable σ that indicates the curthe so-called global embedding principle, see T´ o th (2010); rent operating of thethe construct a lin- on ear model model thatpoint describes thesystem local, and linearized dynamics directly transforming the nonlinear system model via the so-called global embedding principle, see T´ o th (2010); ear that describes local, linearized dynamics Hoffmann and Werner (2015) for an overview, and then rent operating point of thethe system and construct a linandglobal Werner (2015) principle, for an overview, and then ear model describes local, linearized dynamics of the the plantthat around each point. The parameters of the Hoffmann the so-called embedding see T´ o th (2010); Hoffmann and Werner (2015) for an overview, and then of plant around each point. The parameters of the synthesizing an LPV law that gives and ear model that describes dynamics anWerner LPV control control law that gives stability stability and of the plant around each point. Thelinearized parameters of σ resulting model varies withthe σ. local, Next, assuming that σthe is synthesizing Hoffmann and (2015) for an overview, and then synthesizing an LPV control law that gives variations stability and resulting model varies with σ. Next, assuming that is performance guarantees over all possible of of the plant around each point. The parameters of the performance guarantees over all possible variations of resulting model varies with σ. Next, assuming that σ is an external variable (independent from the inputs) an synthesizing an LPV control law that gives variations stability and performance guarantees over all possible of an external variable (independent from the inputs) an σ. Such methods had been thought superior over gainresulting model varies with σ. Next, assuming that σ is σ. Such methods had been thought superior over gainan variable (independent from the inputs) an performance LPVexternal controller dependent on σ σ is is designed designed that, by using using guarantees over all possible variations of σ. Such methods had been thought superior over gainLPV controller dependent on that, by scheduling techniques as they provided direct stability an external (independent from the inputs) an scheduling techniques as they provided directover stability LPV controller dependent on and σ is Packard, designed that, by using linear systemvariable theory (Becker and Packard, 1994), ensures σ. Such methods had been thought superior gainscheduling techniques as they provided direct stability linear system theory (Becker 1994), ensures for the nonlinear system following LPV controller dependent on and σ is Packard, designed that, by using guarantees guarantees for the embedded embedded system following linear system theory (Becker 1994), ensures stability and performance performance specifications for the the LPV model scheduling techniques as theynonlinear provided direct stability guarantees for the embedded nonlinear system following stability and specifications for LPV model aa differential inclusion concept. However, recent studies linear system theory (Becker and Packard, 1994), ensures differential inclusion concept. However, recent studies stability and performance specifications for the LPV model under possible variations of σ in a user specified region of guarantees for the embedded nonlinear system following a differential inclusion concept. However, recent studies under possible variations of σ in a user specified region of Scorletti et al. (2015) indicate that performance issues for stability and performance specifications for the LPV model Scorletti et al. (2015) indicate that performance issues for under possible variations of σ in a user specified region of operating conditions conditions P. P. Finally, Finally, aa nonlinear nonlinear control control law law is is aScorletti differential inclusion concept. However, recentissues studies et al. (2015) indicate that performance for operating reference tracking objectives might still be present. To adunder possible variations of with σ in aameasured user specified region of reference tracking objectives might still be present. To adoperating conditions P. Finally, nonlinear control law is obtained by substituting σ information of Scorletti et al. (2015) indicate that still performance issues for reference tracking objectives might be present. To adobtained by substituting σ with measured information of dress this problem, a strong notion incremental stability operating conditions Finally, nonlinearinformation control law of is dress this tracking problem,objectives a strong notion - incremental stability obtained by substituting σsystem. with ameasured the operating operating point of ofP.the the system. reference might still be present. To has address this problem, a strong notion incremental stability the point was considered, and the corresponding LPV modeling obtained by substituting with measured information of was considered, andathe corresponding LPV modeling has the operating point of theσsystem. dress this problem, strong notion - incremental stability was considered, and the corresponding LPV modeling has a connection to the local linearization of the plant. There are many many approaches available the operating pointapproaches of the system. connection to the local linearization of the plant. There are available to to construct construct an an awas considered, the corresponding LPV a connection to and the local linearization of the modeling plant. has There are many approaches available to construct an LPV model of the plant based on this methodology, see LPV model of theapproaches plant basedavailable on this methodology, see builds stability aContraction connectiontheory to the which local linearization of the plant. results There are many to construct an Contraction theory which builds on on global global stability results LPV model of (2013) the plant based on this methodology, see Bachnas et al. al. (2013) for an an overview. Typically, the plant plant Contraction theory which builds on global stability results Bachnas et for overview. Typically, the from local analysis has gained much attention for nonlinear LPV model of the plant based on this methodology, see from local analysis has gained much attention for nonlinear Bachnas et al. (2013) for an overview. Typically, the plant is linearized linearized around around aa given given set set of of equilibrium equilibrium points points Contraction theoryhas which builds on attention global stability results from local analysis gained much for nonlinear is systems (Lohmiller and Slotine, 1998; Forni and Sepulchre, Bachnas et (2013) for an overview. thepoints plant (Lohmiller andgained Slotine, 1998; Forni and Sepulchre, is linearized a given set set of Typically, equilibrium (griding of al. P)around and the the resulting set of LTI models models are systems from local analysis has much attention for nonlinear systems (Lohmiller and Slotine, 1998; Forni and Sepulchre, (griding of P) and resulting of LTI are 2014). Related works include velocity linearization (Leith is linearized around given set set ofis of equilibrium points Related works velocity linearization (Leith (griding and resulting LTI models are 2014). of P) interpolated over P the ora linearization linearization accomplished over systems (Lohmiller andinclude Slotine, 1998; Forni and Sepulchre, 2014). Related works include linearization (Leith interpolated over P or is of accomplished over and Leithead, 2000) and Gˆ aavelocity teaux derivative (Fromion (griding of P) and the resulting set LTI models are and Leithead, 2000) and Gˆ teaux derivative (Fromion interpolated over P or linearization is accomplished over input and state trajectories. Similarly, the LPV controller 2014). Related works include velocity linearization (Leith and Leithead, 2000) and Gˆ a teaux derivative (Fromion input and state trajectories. Similarly, the LPV controller et al., 2001; Fromion and Scorletti, 2003). Recently, coninterpolated overtrajectories. Pdesigning or linearization is the accomplished over et al.,Leithead, 2001; Fromion and Scorletti, 2003). Recently, coninput and state Similarly, LPV controller can be obtained by LTI controllers separately for and 2000) and Gˆ a teaux derivative (Fromion et al., 2001; Fromion and Scorletti, 2003). Recently, concan be obtained by designing LTI controllers separately for traction analysis was extended to constructive nonlinear input and state trajectories. Similarly, the LPV controller traction analysis was extended to constructive nonlinear can be obtained by designing LTI controllers separately for finite set of values of σ and then interpolating these LTI et al., 2001; Fromion and 2003). Recently, contraction analysis extended to constructive nonlinear finitebeset of values of σ and then interpolating these LTI control design bywas using aScorletti, differential version of control can obtained by designing LTI controllers separately for control design by using a differential version of control finite set of values of σ and then interpolating these LTI traction analysis was extended to constructive nonlinear control design by using a differential version of control Lyapunov function Control Contraction Metric (CCM) work supported by the Australian Research Council. finite of was values of σ and then interpolating these LTI Lyapunov function - Control Contraction Metric This This set work was supported by the Australian Research Council. control design bySlotine, using a2017, differential version of (CCM) control Lyapunov function - Control Contraction Metric (CCM) (Manchester and 2018). Further extensions This work has received funding European Research Council This work was supported byfrom thethe Australian Research Council. (Manchester and Slotine, 2017, 2018). Further extensions This work has received funding from the European Research Council Lyapunov function Control Contraction Metric (CCM) (Manchester and Slotine, 2017,include 2018). distributed Further extensions of the approach control This work thethe Australian Research Council. (ERC) under the supported European Unions Horizon 2020 research and This work haswas received fundingby from European Research Council of the CCM CCM based based approach control (ERC) under the European Unions Horizon 2020 research and (Manchester Slotine, 2017,include 2018). distributed Further extensions of the CCM and based approach include distributed control This work has received funding from theHorizon European Research Council innovation programme (grant agreement nr. 714663). (ERC) under the European Unions 2020 research and innovation programme (grant agreement nr. 714663). of the CCM based approach include distributed control (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement nr. 714663). innovation programme agreement nr. 714663). 2405-8963 © 2019, IFAC(grant (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.346
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
(Shiromoto et al., 2018) and distributed economic model predictive control (MPC) (Wang et al., 2017). The main contribution of this paper is a comparison study between the CCM based nonlinear control approach and the LPV gain scheduling technique using local linearization. For simplicity, only state-feedback control design is considered. We show that CCM based control is an extended LPV gain scheduling approach. First, the so-called differential dynamics in contraction theory can be seen as a local LPV model which takes linearization along any admissible solution rather than an equilibrium family in conventional gain-scheduling. Second, similar parameterdependent linear matrix inequality (LMI) conditions are derived as in local LPV synthesis. One difference is that the CCM based approach explicitly takes the original nonlinear plant into account leading to less conservative results. Furthermore, the control realization integrates a series of local controllers on a particular path joining the current and reference state trajectory, which leads to an exact realization without any hidden coupling term. Based on this, local stability and performance design can be carried onto the entire state space as the length of the path shrinks exponentially. Paper outline. Section 2 presents formulations of different stability and performance. Section 3 gives a brief review of the LPV gain scheduling approach using local linearization, which is mainly adopted from Rugh and Shamma (2000). Section 4 discusses the various connections and extensions between CCM and LPV based approaches. An illustrative example is presented in Section 5. 2. PRELIMINARIES Let |x| be the Euclidean norm of a vector x. For any matrix A ∈ Rn×n , we use the notation He{A} := A + A . Positive (negative) definiteness of a Hermitian matrix X is denoted as X 0 (X ≺ 0). C k denotes the set of vector signals on R which are k th times differentiable. L2 is the space of square-integrable vector signals on R≥0 , i.e., ∞ |f (t)|2 dt < ∞. The causal truncation (·)T is f := 0 defined by (f )T (t) := f (t) for t ≤ T and 0 otherwise. Le2 is the space of vector signals on R≥0 whose causal truncation belongs to L2 .
In this paper, we consider a nonlinear system x˙ = f (x, u, w), z = h(x, u, w) (1) nx nu nw where x(t) ∈ R , u(t) ∈ R , w(t) ∈ R , z(t) ∈ Rnz are state, control, external input and performance output signals at time t ∈ R≥0 , respectively. The functions f and h are assumed to be smooth and time-invariant. We define a target trajectory to be a forward-complete solution of (1), i.e., a pair (x∗ , u∗ , w∗ , z ∗ )(·) with x∗ (·) piecewise differentiable and (u∗ , w∗ , z ∗ )(·) piecewise continuous satisfying (1) for all t ∈ R≥0 . The target trajectory is said to be an equilibrium if (x∗ , u∗ , w∗ , z ∗ )(·) = (xe , ue , we , ze ). For simplicity, we assume that the nominal external input is w∗ (·) = 0. We will consider state-feedback controllers of the form u(t) = κ(x(t), x∗ (t), u∗ (t)). (2) To define a nonlinear control problem precisely, we must be specific about the notion of stability and performance.
45
The closed-loop system of (1) and (2) is said to be globally asymptotically stable with respect to the target trajectory (x∗ , u∗ )(·) if the closed-loop solution x(t) of the nominal system (i.e., w = w∗ ) exists and satisfies 1) For any there exists an ρ such that |x(0)−x∗ (0)| < ρ implies |x(t) − x∗ (t)| < , 2) For any initial condition x(0) ∈ Rn , the closed-loop solution satisfies |x(t) − x∗ (t)| → 0.
Global exponential stability is a stronger notion and requires that there exists a R and λ such that |x(t) − x∗ (t)| ≤ Re−λt |x(0) − x∗ (0)|, ∀x(0) ∈ Rn . (3) The closed-loop system is said to achieve L2 -gain performance of α for the target trajectory (x∗ , u∗ , w∗ , z ∗ )(·), if for any initial condition x(0) and input w such that w − w∗ ∈ Le2 , and for all T > 0 solutions exist and satisfy (4) (z − z ∗ )T 2 ≤ α2 (w − w∗ )T + β(x(0), x∗ (0)) where β(x1 , x2 ) ≥ 0 with β(x, x) = 0.
The above reference-related stability and performance notions allow us to investigate different formulations of control objectives in a unified way. Let B∗ be the set of target trajectories which are used for the definitions of asymptotic (exponential) stability and L2 performance. In a standard formulation, B∗ only contains one “preferred” trajectory – the zero solution (i.e., f (0, 0, 0) = 0, h(0, 0, 0) = 0). A stronger formulation, called equilibrium-independent asymptotic (exponential) stability and L2 gain, is referred to the case where B∗ is chosen to be the set of all possible equilibrium points (Simpson-Porco, 2019). The socalled universal exponential stability and L2 gain is an even stronger formulation which requires B∗ to include all admissible trajectories of the nominal system (Manchester and Slotine (2017, 2018)). The incremental formulation is referred to the case where asymptotic (exponential) stability and L2 gain are satisfied for any pair of system trajectories. Note that universal exponential stability is equivalent to incremental exponential stability. But universal L2 gain is weaker than the incremental one (Manchester and Slotine, 2018). 3. GAIN SCHEDULING APPROACH 3.1 System Linearization To construct an LPV representation for (1) using local modelling concept, we assume that the equilibrium points are uniquely characterized by xe . Hence, to describe the equilibrium points associated local dynamics of the system, we introduce a scheduling variable σ ∈ Rnσ that depends on the state, i.e., σ = g(x) (5) where g is a smooth vector function. Note that σ can also depend on w if it is measurable. Here the possible trajectories of the scheduling signal σ(t) are assumed to belong to the set ˙ ∀t ≥ 0} ˙ ∈ P, (6) T = {σ ∈ C 1 : σ(t) ∈ P, σ(t) nσ nσ ˙ where P = {σ ∈ R : |σi | ≤ σ i } and P = {p ∈ R : |pi | ≤ pi } with 1 ≤ i ≤ nσ . Using σ, the equilibrium family is characterized as the set {(xe , ue , ze , we )(σ)}σ∈P where xe (·), ue (·), ze (·), we (·) are smooth functions, and (xe , ue , ze , we )(σ) is an equilibrium of (1) for all σ ∈ P.
46
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
By linearizing (1) around the equilibrium family, we obtain an LPV model as follows: xδ A(σ) Bu (σ) Bw (σ) x˙ δ uδ , σ ∈ P (7) = C(σ) Du (σ) Dw (σ) zδ wδ where xδ = x − xe (σ), uδ = u − ue (σ), wδ = w − we (σ), zδ = z − ze (σ) are deviation variables. The matrices A, Bu , Bw , C, Du , Dw are defined as the evaluations of ∂f ∂f ∂f ∂h ∂h ∂h ∂x , ∂u , ∂w , ∂x , ∂u , ∂w at the σ defined equilibrium point.
σ( x e(
t))
xe (σ)
Fig. 1. Illustration of gain-scheduling control.
3.2 Control Synthesis Consider the static LPV controller of the form uδ = K(σ)xδ , which yields a closed-loop LPV system x˙ δ A(σ) B(σ) xδ = C(σ) D(σ) wδ zδ
x(t)
(8)
(9)
with A = A + Bu K, B = Bw , C = C + Du K, D = Dw . Theorem 1. The unforced closed-loop system (10) x˙ δ = A(σ)xδ , σ ∈ T is exponentially stable if there exists a M (σ) 0 such that nσ ∂M (σ) He{M (σ)A(σ)} + 2λM (σ) + ρi ≺ 0 (11) ∂σi i=1 ˙ for all σ ∈ P and ρ ∈ P.
The above theorem implies that V (xδ , σ) := x δ M (σ)xδ is a parameter-dependent Lyapunov function for system (10). By applying a congruence transformation (12) W (σ) = M −1 (σ), L(σ) = K(σ)W (σ), we can obtain a convex formulation: nσ ∂W (σ) He{A(σ)W (σ)+Bu (σ)L(σ)}+2λW (σ)− ρi ≺0 ∂σi i=1 (13) ˙ for all σ ∈ P and ρ ∈ P.
Note that due to linearity of the LPV description (9), exponential stability and L2 gain bound of (9) w.r.t. the origin is equivalent to the equilibrium-independent asymptotic stability and L2 gain (Rugh and Shamma, 2000; Hoffmann and Werner, 2015). Theorem 2. A controller (8) achieves an L2 performance level of α for LPV system (7) if there exists M (σ) 0 ˙ such that, for all σ ∈ P and ρ ∈ P, −1 M(σ, ρ) M (σ)B(σ) α C (σ) B (σ)M (σ) (14) −I α−1 D (σ) ≺ 0 −1 −1 α C(σ) α D(σ) −I nσ (σ) ρi ∂M where M(σ, ρ) = He{M (σ)A(σ)} + i=1 ∂σi . Based on the above condition, we can synthesize an LPV controller which achieves the minimal L2 -gain bound for the closed-loop LPV system. 3.3 Controller Realization The LPV control realization problem is to construct a gain-scheduled law u = κ(x, σ) such that
ue (σ) = κ(xe (σ), σ), (15a) ∂κ (xe (σ), σ) = K(σ). (15b) ∂x Condition (15b) implies that linearization of u = κ(x, σ) at this equilibrium is the LPV controller (8). An intuitive choice of control realization in the literature is u = ue (σ) + K(σ)[x − xe (σ)]. (16) Under the assumption that the equilibrium points of (1) are uniquely characterized by xe , σ can be expressed in terms of x via (5). Using this relation, (16) reads as u = ue (g(x)) + K(g(x))[x − xe (g(x))]. (17) The main “trick” behind of this gain-scheduling approach is that σ is treated as a parametric/dynamic uncertainty throughout the design process, but during controller realization is substituted by a function of a measured variable characterizing the operating point changes (Rugh and Shamma (2000)). Although σ is implicitly involved via equilibrium parameterizations, linearization of (17) may not satisfy condition (15b) since ∂xe (σ) ∂g ∂ue (σ) uδ = K(σ)xδ + − K(σ) (xe (σ))xδ ∂σ ∂σ ∂x contains additional terms, called hidden coupling terms, compared with (8). These terms may lead to closed-loop instability regardless the fact that exponential stability is achieved in the control synthesis stage, which is a well-known drawback of the local LPV controller (see Example 8 in Rugh and Shamma (2000)). 3.4 Stability and Performance Assessment The core idea of gain-scheduled control (17) is to track a reference xe (σ(t)) lying on the equilibrium manifold, as shown in Fig. 1. This strategy achieves local equilibriumindependent stability if the schedule signal σ(t) is sufficiently “slowly varying” (Rugh and Shamma, 2000). The main reason is that the scheduled reference trajectory (xe , ue , we , ze )(σ(t)) is not admissible to the closed-loop system x˙ = f (x, κ(x, σ), w) with σ = g(x) since simple substitution yields a residual term E(σ)σ(t) ˙ with E(σ) = ∂xe (σ) . Therefore, the actual linearization of the closed∂σ loop system with wδ (t) = 0 is x˙ δ = A(σ)xδ − E(σ)σ. ˙ (18) If the rates of parameter variation are not “sufficiently slow”, the residual terms can drive the state away from the small neighborhood of xe (σ), which may violate the local stability design. To ensure global stability and performance, excessive simulations or even experiments are needed.
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
4. CCM BASED NONLINEAR CONTROL In this section, we will show the connections and differences between the CCM-based control and local LPVbased gain-scheduled control. Both approaches use very close LPV modeling and control synthesis formulations. The major differences come from the ways to interpret and use the LPV tools. In the CCM-based approach, LPV modeling, design and realization are carried out for the entire state space, which is an extension to the local gainscheduling approach as it only considers the equilibrium manifold (as shown in Fig. 1). With this conceptual innovation, the CCM-based approach can provide an exact control realization which does not contain any hidden coupling term and achieve universal stability and L2 performance.
Firstly, we recall some basic facts of Riemannian geometry from Do Carmo (1992). A Riemannian metric on Rn is a smooth matrix function M (x) 0 which defines an inner product δ1 , δ2 x = δ1 M (x)δ2 for any two tangent vector δ1 , δ2 . A metric is called uniformly bounded if there exist positive constants a2 ≥ a1 such that a1 I ≺ M (x) ≺ a2 I, ∀x ∈ Rn . Γ(x0 , x1 ) denotes the set of piecewise smooth paths c : [0, 1] → Rn with c(0) = x0 and c(1) = x1 . The curved length and energy of c(·) is defined by 1 1 cs , cs c(s) ds and ε(c) = cs , cs c(s) ds (c) = 0
0
where cs = ∂c/∂s, respectively. The geodesic γ(·) denotes a path with the minimal length, i.e., (γ) = inf c∈Γ(x0 ,x1 ) (c). The Riemann distance and energy between x0 and x1 are defined by d(x0 , x1 ) = (γ) and ε(x0 , x1 ) = ε(γ) = 2 (γ).
4.1 System Linearization By choosing σ = (x, u, w), we can construct a continuously linearized system (so-called differential dynamics) δ δ˙x = A(σ) Bu (σ) Bw (σ) δx (19) u C(σ) Du (σ) Dw (σ) δz δw where the matrices A, Bu , Bw , C, Du , Dw are defined in a similar way as in the local gain-scheduling approach. The variables δx , δu , δw , δz are the virtual displacement between neighboring solutions (Lohmiller and Slotine, 1998) or the tangent vector of the solution manifold (Forni and Sepulchre, 2014). Other related linearization techniques include velocity linearization (Leith and Leithead, 2000) and Gˆ ateaux derivative (Fromion et al., 2001). Remark 3. Note that the differential dynamics (19) can be seen as a local LPV system defined on the entire state space rather than the equilibrium manifold. 4.2 Control Synthesis The control synthesis searches for a differential controller: (20) δu = K(σ)δx := K(x, u)δx which stabilize the unforced closed-loop dynamics (21) δ˙x = A(σ)δx := [A(σ) + Bu (σ)K(σ)]δx .
It can be achieved by a sufficient condition as follows nx ∂M (x) fi (σ) ≺ 0 (22) He{M (x)A(σ)} + 2λM (x) + ∂xi i=1
47
where fi is the model of the state xi . The above synthesis formulation is very close to (11). Thus, similar convexation technique can be applied here. The main difference between these two formulation is that (22) uses detailed model (1) to describe the scheduling signal σ(t) while (13) uses coarse description (6) which only contains the region of the parameter and its variation. This can lead to less conservative results. For instance, a non-uniform metric M (x) can be found even if both the parameter σ and its variation σ˙ are unbounded, e.g., the system x˙ = −x − x3 admits a non-uniform metric M (x) = 1 + 3x2 . Here V (x, δx ) = δx M (x)δx is called control contraction metric (CCM), which is a differential control Lyapunov function validated everywhere in the state space. Since the schedule variable σ in the gain scheduling approach only depends on x, the control Lyapunov matrix M (σ) obtained from (11) can be seen as a CCM defined on the equilibrium manifold. For performance analysis, we can obtain a formulation similar to (14): M M B α−1 C B M −I α−1 D ≺ 0 (23) α−1 C α−1 D −I nx (x) fi (σ) ∂M where M = He{M (x)A(σ)} + i=1 ∂xi . 4.3 Controller Realization
As discussed in Section 3.3, the differential controller (20) is generally not completely integrable, i.e., there does not exist a gain-scheduled law κ(·) whose Jacobian is K. Unlike the LPV control, the CCM based approach only considers a much weaker condition - the path-integrability of K. Let γ be a geodesic (a minimal path) joining x∗ (t) and x(t), which can be obtained by solving a simple model predictive control problem online (Leung and Manchester, 2017). The state-feedback law is the integral of the differential control (20) over the path γ: (24) u(t) = κ(x(t), x∗ (t)) := κγ (t, 1) where κγ (t, s) is the unique solution of the following integral equation s ∗ κγ (t, s) := u (t) + K(γ(s), κγ (t, s))γs (s)ds. (25) 0
Remark 4. Note that κγ (t, s) satisfies κγ (t, 0) = u∗ and ∂κγ (t, s) = K(γ(s), κγ (t, s))γs (s), ∀s ∈ [0, 1]. ∂s Thus, it does not contain any hidden coupling term and serves as an exact realization for the differential controller (20). Moreover, it can be also applied to those approaches using incremental analysis (Fromion and Scorletti, 2003) where exact realization for general nonlinear systems is an open problem (Scorletti et al., 2015).
We give a geometric interpretation about the interconnection between the path of control signal κγ (t, s) and the local LPV controller (16). Let 0 = s0 < s1 < · · · < sN = 1 with sj+1 − sj be sufficiently small. For any frozen time t, the integral equation (25) gives κγ (sj+1 ) ≈ κγ (sj ) + K(γ(sj ), κγ (sj ), w)[γ(sj+1 ) − γ(sj )],
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
48
x(
γs sj+1
t)
γ(s)
sj x∗ (t)
Fig. 2. Geometric illustration of the control realization in CCM based approach. where the argument t is omitted for simplicity. Thus, κγ (sj+1 ) is an LPV controller (16) that stabilizes the state γ(sj+1 ) around γ(sj ), as shown in Fig. 2. Based on this observation, κγ integrates a series of local LPV controllers (20) along a particular path γ and the CCM based gain scheduling law (24) is the corresponding control action to the measured state x(t) = γ(t, 1). 4.4 Stability and Performance Assessment Since the control realization (25) is exact, the differential dynamics of the closed-loop system along the path γ is same as (21), which is exponentially stable: d (δ M (x)δx ) = δx M˙ δx + 2δx M Aδx ≤ −2λδx M (x)δx . dt x The global stability follows by integrating the above inequality along a geodesic γ: d ε(x∗ (t), x(t)) ≤ −2λε(x∗ (t), x(t)). (26) dt An explanation using LPV concepts is given as follows. The smooth path γ can be understood as a “chain” of many states joining the current state x(t) to the reference point x∗ (t), and the tangent vector δx = γs as a “link” whose behavior is described by the closed-loop differential dynamics (21), as shown in Fig. 2. The convergence of x(t) to x∗ (t) can be inferred since each link in the chain gets shorter due to local stability. The following theorems give the global stability and performance results for the CCM based approach. Theorem 5. (Manchester and Slotine (2017)). If there exists a uniformly bounded metric a1 I ≤ M (x) ≤ a2 I for which (22) holds for all x, u, w, then system (1) under the controller (24) is universal exponentially stable with rate a2 λ and overshoot C = a1 . Theorem 6. (Manchester and Slotine (2018)). The closedloop system of (1) and (24) has an universally L2 -gain bound α if there exists a uniformly bounded metric M (x) such that (23) holds for all x, u, w. 5. CASE STUDY Consider the following nonlinear system −x1 − x2 + w x˙ 1 = (27) x˙ 2 1 − e−x2 + u where w is a measurable reference signal. This example was used in Rugh (1991) to illustrate gain scheduling design for nonlinear systems. Assume that the control task is to stabilize the system at the equilibrium family 0 xe = , ue = e−we − 1. (28) we
In this section, we compare tracking control of (27) with time-varying we using LPV and CCM-based approaches. For LPV design, we introduce the scheduling as σ = e−we which in terms of the equilibrium point relation is equivalent with σ = e−x2,e . We can obtain an LPV model of the system with coefficient matrices −1 −1 0 A(σ) = , B(σ) = . (29) 0 σ 1
To derive a gain-scheduling controller, we consider placing both the closed-loop eigenvalues at −2, leading to K(σ) = [1 −3 − σ] . (30) Then, the control law (16) with the choice of σ = e−w corresponds to the nonlinear controller u = ue (w) + x1 − (3 + e−w )(x2 − w) (31) which is referred as Gain-Scheduled Controller (GSC) 1. The differential dynamics of the closed-loop system can be represented by −1 −1 δ˙x = A(x)δx = (32) δ 1 a(x2 , w) x where a(x2 , w) = e−x2 − e−w − 3. Since A(x) has positive eigenvalues if x2 < − ln(4 + e−w ), the closed-loop system is unstable in this region.
By implementing the scheduling law according to the equilibrium relation σ = e−x2 , (17) gives the controller GSC 2 as follows u = x1 + e−x2 − 1. (33) Linearization of the closed-loop system at the reference xe (w(t)) = [0, w(t)] yields −1 −1 0 x˙ δ = xδ − . (34) 1 0 w˙
The closed-loop system is globally exponential stable. But the √ differential dynamics have eigenvalues λ1,2 = −1/2 ± 3/2i with larger real parts than the specified ones λ1,2 = −2. This mismatch is caused by the hidden coupling terms: ∂xe (σ) ∂κ(xe (σ), σ) ∆ ∂ue (σ) Kh (σ) = −K(σ) − = 3. (35) ∂σ ∂σ ∂σ For CCM based control design, we choose the following differential state feedback control δu = K(x)δx (36) with K(x) = 1 −(3 + e−x2 ) . This leads to exponentially stable closed-loop differential dynamics with the same eigenvalues as the LPV controller. Thus, we can obtain a constant CCM for the closed-loop system, which implies that the geodesic between x∗ and x is a straight line (i.e., γ(s) = (1 − s)x∗ + sx). Further, the CCM controller can be computed as 1 K(γ(s))(x − x∗ )ds u = u∗ + (37) 0 = x1 + e−x2 − 1 − 3(x2 − w) + w˙ where the target trajectory is x∗ (t) = [0, w(t)] , u∗ (t) = ˙ The closed-loop system e−w(t) − 1 + w(t). d x1 −1 −1 x1 = (38) 1 −3 x2 − w dt x2 − w is globally exponential stable at x∗ (t).
Ruigang Wang et al. / IFAC PapersOnLine 52-28 (2019) 44–49
Piecewise constant reference
x2
5
0 Ref GSC 1 GSC 2 CCM
-5
0
5
10
15
20
25
Time-varying reference
x2
5
0 Ref GSC 1 GSC 2 CCM
-5
0
5
10
15
time
Fig. 3. Closed-loop trajectories for piecewise-constant and time-varying reference. For comparison study, we consider two tracking scenarios: piecewise-constant setpoints and time-varying references. As shown in Fig. 3, the closed-loop system under GSC 1 is not stable when the state x2 enters into certain regions. Although the controller GSC 2 can ensure equilibriumindependent stability, the transitions are much slower and contain oscillations, compared with other controllers. This is caused by the hidden coupling terms in (35). For timevarying references, GSC 2 cannot reach zero error due to the term w˙ in (34). The CCM based approach overcomes these issues and achieves universal stability. 6. CONCLUSION In this paper, we investigated the apparent connection between contraction theory based nonlinear controller design and the gain-scheduling approach which corresponds to LPV control based on local linearization of the nonlinear system. We show that the CCM based control is an extended LPV gain scheduling approach as it yields a control realization without any hidden coupling term and achieves universal stability. REFERENCES Bachnas, A.A., T´ oth, R., Mesbah, A., and Ludlage, J. (2013). A review on data-driven linear parametervarying modeling approaches: A high-purity distillation column case study. J. Process Control, 24(4), 272–285. Becker, G. and Packard, A. (1994). Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Syst. Control Lett., 23(3), 205–215. Do Carmo, M.P. (1992). Riemannian Geometry. Springer, Bosten, USA.
49
Forni, F. and Sepulchre, R. (2014). A differential lyapunov framework for contraction analysis. IEEE Trans. Autom. Control, 3(59), 614–628. Fromion, V., Monaco, S., and Normand-Cyrot, D. (2001). The weighted incremental norm approach: from linear to nonlinear H∞ control. Automatica, 37(10), 1585–1592. Fromion, V. and Scorletti, G. (2003). A theoretical framework for gain scheduling. Int. J. Robust Nonlin. Control, 13(10), 951–982. Hoffmann, C. and Werner, H. (2015). A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations. IEEE Trans. on Control Syst. Techno., 23(2), 416–433. Klatt, K.U. and Engell, S. (1998). Gain-scheduling trajectory control of a continuous stirred tank reactor. Comput. & Chem. Eng., 22(4), 491–502. Leith, D.J. and Leithead, W.E. (2000). Survey of gainscheduling analysis and design. Int. J. Control, 73(11), 1001–1025. Leung, K. and Manchester, I.R. (2017). Nonlinear stabilization via control contraction metrics: A pseudospectral approach for computing geodesics. In Proc. Amer. Control Conf. (ACC), 1284–1289. Seattle, WA. Lohmiller, W. and Slotine, J.J.E. (1998). On contraction analysis for non-linear systems. Automatica, 34(6), 683– 696. Manchester, I.R. and Slotine, J.J.E. (2017). Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design. IEEE Trans. Autom. Control, 62(6), 3046–3053. Manchester, I.R. and Slotine, J.J.E. (2018). Robust control contraction metrics: A convex approach to nonlinear state-feedback H∞ control. IEEE Control Syst. Lett., 2(3), 333–338. Papageorgiou, G., Glover, K., D’Mello, G., and Patel, Y. (2000). Taking robust LPV control into flight on the VAAC Harrier. In Proc. Conf. on Decision and Control (CDC), 4558–4564. Sydney, Australia. Rugh, W.J. (1991). Analytical framework for gain scheduling. IEEE Control Syst. Mag., 11, 74–84. Rugh, W.J. and Shamma, J.S. (2000). Research on gain scheduling. Automatica, 36, 1401–1425. Scorletti, G., Fromion, V., and De Hillerin, S. (2015). Toward nonlinear tracking and rejection using LPV control. In Proc. IFAC Workshop on Linear Parameter Varying Systems, 13–18. Grenoble, France. Shiromoto, H.S., Revay, M., and Manchester, I.R. (2018). Distributed nonlinear control design using separable control contraction metrics. IEEE Trans. Control Netw. Syst. doi:10.1109/TCNS.2018.2885270. Simpson-Porco, J.W. (2019). Equilibrium-independent dissipativity with quadratic supply rates. IEEE Trans. Autom. Control, 64(4), 1440–1455. T´oth, R. (2010). Modeling and Identification of Linear Parameter-Varying Systems. Springer, Heidelberg. Wang, R., Manchester, I.R., and Bao, J. (2017). Distributed economic MPC with separable control contraction metrics. IEEE Control Syst. Lett., 1(1), 104–109.