2.4 Linear Parameter Varying Modelling
2.4.1 Discrete-time LPV systems
¤
𝑥 =cos(𝜌)𝑣
¤
𝑦 =sin(𝜌)𝑣 𝜌 ∈ [−𝜋 𝜋] 𝜃¤=𝜔
however, the resulting qLPV model is not useful for control, since the model lost all coupling information between the states𝑥 , 𝑦and𝜃; furthermore this model is not stabilizable for the whole operating regiona. A Jacobian linearization, on the other hand would result in𝜌
1=𝜃, 𝜌2 =𝑣
Σ2
¤˜
𝑥 ≈ cos(𝜌
1)𝑣˜−sin(𝜌
1)𝜌
2𝜃˜ 𝜌
1 ∈ [−𝜋 𝜋]
¤˜
𝑦≈ sin(𝜌
1)𝑣˜+cos(𝜌
1)𝜌
2𝜃˜ 𝜌
2 ∈ [−1 1]
¤˜ 𝜃 ≈ 𝜔˜
which, at the price of being an approximate model and having more parameters, it is far more meaningful in the context of control. Note that this model also becomes uncontrollable for(𝜌
1, 𝜌
2) =(0,0),(𝜌
1, 𝜌
2) =(𝜋,0)however this comes from the underlying system, and not because of artifacts introduced by parameterization, as in the previous case.
athis is an important assumption usually made in order to use Lyapunov arguments to establish stability and synthesize controllers
Beside the two presented parameterization methods, others exist which may be better suited for certain applications. In Chapter 4 a different method is presented based on velocity-based linearization [73], this approach offers the advantage that it is exact as ad-hoc parameterization but as easy to obtain and coupling-preserving as the Jacobian linearization method.
2.4.1 Discrete-time LPV systems
Even though most of the design and synthesis results in the LPV literature focus on the continuous-time setting, it is of utmost practical interest to consider a discrete-time setting as well. This is perhaps even more important in the present context as predictive control laws are considerably more easily handled in discrete-time. With this in mind, the definitions above can be appropriately modified to accommodate the effect of sampling.
Definition 2.17 (Discrete-time LPV system). A discrete-time Linear Parameter Varying (LPV) system is a dynamic system of the form
𝑥𝑘+
1= 𝐴(𝜌𝑘)𝑥𝑘+𝐵(𝜌𝑘)𝑢𝑘
𝑦𝑘 =𝐶(𝜌𝑘)𝑥𝑘 +𝐷(𝜌𝑘)𝑢𝑘 (2.21) where 𝜌 ∈ R𝑛𝜌 is an unknown but measurable time-varying parameter. The parameter vector𝜌 belongs to a compact set of admissible values, i.e.
𝜌𝑘 ∈ P ∀𝑘 ≥ 0.
The maps 𝐴 : R𝑛𝜌 → R𝑛×𝑛, 𝐵 : R𝑛𝜌 → R𝑛×𝑚, 𝐶 : R𝑛𝜌 → R𝑙×𝑛, 𝐷 : R𝑛𝜌 → R𝑙×𝑚 are continuous functions of 𝜌onP.
In addition, the parameterdifferencecan also be bounded, so as to belong to a compact set V
Δ𝜌𝑘 ∈ V, ∀𝑘 ≥ 0.
The sets of admissible trajectories FP, FPV can be appropriately redefined with the difference that 𝜌𝑘 ∈ C1(Z≥0,R𝑛𝜌). Finally discrete-time quasi-LPV systems can be defined as
Definition 2.18(quasi-LPV system). A quasi-LPV system is an LPV system as defined in Definition 2.17 for which the time varying parameters are functions of endogenous signals, i.e.
𝜌𝑘 = 𝜚(𝑥𝑘, 𝑢𝑘). where 𝜚:R𝑛×R𝑚 →R𝑛𝜌 is a continuous function onP. Input-Output LPV models
As in the LTI case, an equivalent representation of an LPV system in the form (2.21) can be obtained which only depends on time-shifted inputs and outputs. Although such a representation arises more naturally when performing system identification using the prediction error method [14], first principle models can also be derived directly e.g. by appropriate discretization of the nonlinear differential equations.
Definition 2.19 (Input-Output LPV model). A Input-Output Linear Parameter Varying (LPV) model is a model of the form
𝑦(𝑘) =
𝑛𝑑 𝑦
Õ
𝑖=1
𝐴𝑖(𝜌(𝑘))𝑞−𝑖𝑦(𝑘) +
𝑛𝑑𝑢
Õ
𝑗=1
𝐵𝑗(𝜌(𝑘))𝑞−𝑗𝑢(𝑘) (2.22) where𝑦 ∈R𝑙 is the output,𝑢 ∈R𝑚 is the input and𝜌 ∈R𝑛𝜌is an unknown but measurable time-varying parameter. The parameter vector 𝜌 belongs to a compact set of admissible values, i.e.
𝜌𝑘 ∈ P ∀𝑘 ≥ 0. The maps 𝐴𝑖 : R𝑛𝜌 → R𝑙×𝑙 𝑖 = 1, ..., 𝑛𝑑
𝑦, 𝐵𝑗 : R𝑛𝜌 → R𝑙×𝑚 𝑗 = 1, ..., 𝑛𝑑
𝑢 are continuous functions of𝜌 onP.
In addition, the parameterdifferencecan also be bounded, so as to belong to a compact set V
Δ𝜌𝑘 ∈ V, ∀𝑘 ≥ 0.
Correspondingly, in the quasi-LPV case, the parameters are now functions of inputs and outputs.
Definition 2.20(quasi-LPV IO model). An IO quasi-LPV model is an LPV model as defined in Definition 2.19 for which the time varying parameters are functions of endogenous signals, i.e.
𝜌𝑘 = 𝜚(𝑦𝑘, 𝑢𝑘). where 𝜚:R𝑙×R𝑚 → R𝑛𝜌is a continuous function on P.
A more detailed discussion of IO qLPV representations is given in Chapter 5 in the context of predictive control for IO quasi-LPV systems.
Dynamic Parameter Dependence
An issue with discrete-time LPV models which is excluded from the definitions above, is the possibility to exhibit dependence on time-shifted parameter values. This is an undesired effect often stemming from conversions between IO and state space (SS) models.
Definition 2.21(Dynamic Parameter Dependence). An LPV model is said to exhibit dy-namic parameter dependence if the model matrices are function of current, as well as time-shifted, parameter values, i.e. if (𝐴(P𝑘), 𝐵(P𝑘), 𝐶(P𝑘), 𝐷(P𝑘)) in the SS case and 𝐴𝑖(P𝑘), 𝐵𝑗(P𝑘),𝑖 =1, ..., 𝑛𝑑
𝑦, 𝑗 =1, ..., 𝑛𝑑
𝑢 in the IO case where P(𝑘) =
𝜌(𝑘) 𝜌(𝑘+𝑛+
1) ... 𝜌(𝑘−𝑛−
1) ...
where𝑛+
𝑖, 𝑛−
𝑖 ∈Z≥0, if in the other hand, the matrices are functions of only current parameter values 𝜌𝑘, the model is said to have static parameter dependence.
The definition for the LPV models above only consider the static parameter dependence case, this although less general, represent the models that are of interest in the context of this thesis, unless otherwise stated.
MPC for quasi-LPV Systems: State Space Framework
This chapter presents the framework on which the rest of this work is based. The basic premise is to use quasi-LPV modelling as a tool to embed nonlinear system dynamics in a linear model;
this allows the optimization problem which arises in MPC to be efficiently solved. Furthermore, this mature modelling framework enables the use of several available results that prove useful when establishing stability conditions; as LPV systems naturally extend many non-predictive LTI stability results to nonlinear systems, so can similar procedures be applied in the context of MPC. Whereas for gain-scheduling control it is irrelevant whether the scheduling parameters are exogenous or endogenous, in the context of MPC the former case is such that evolution of the scheduling variables is unknown and can thus not be predicted; for this reason, MPC for LPV systems faces the problem that the prediction model is uncertain. In the case of quasi-LPV systems, the scheduling variable is determined by the state and/or input and can therefore be predicted, although the resulting prediction equations are nonlinear. To carry out this prediction efficiently, an iterative scheme is proposed which is obtained by freezing the scheduling trajectory to the one obtained in the previous time-step, making the prediction linear time-varying, thus enabling efficient optimization.
Stability analysis is based on the concept ofdual mode control[85], the infinite horizon prediction is made assuming that a region in state space containing the origin is reached within a finite horizon, control authority is then switched to fixed state feedback control law which, within this region, generates feasible control trajectories to drive the system to the origin. The novelty of the proposed approach lies in extending analysis to parameter dependent terminal region and controller, thus making them less conservative and comparatively easier to compute than those obtained by other methods.
This chapter is organized as follows: Section 3.1 gives the problem setup and presents the parameter dependent predictions along with an iterative algorithm which can be used to solve the resulting online optimization problem efficiently. The stability result is given in Section 3.2 where stability conditions in the form of LMIs, to be solved offline, are derived. Section 3.3 discusses the extension to the tracking problem and how offset-free tracking can be achieved, while Section 3.4 presents how to consider nonlinear constraints in this framework. Finally, Section 3.5 validates the control scheme experimentally on an Arm-Driven Inverted Pendulum and Section 3.6 summarizes the chapter.