Applied LPV Control with Full Block Multipliers and Regional Pole Placement

Abstract

A formulation of an LPV control problem with regional pole placement constraints is presented, which is suitable for the application of a Full Block S-Procedure. It is demonstrated that improved bounds can be obtained on the induced 𝐿₂ norm of closed loop systems, while satisfying pole placement constraints. An application consisting in the 6 degrees of freedom (DOF) control of a space vehicle is developed as an example, with hardware in the loop (HIL) simulation. This shows that the method is appealing from the practical point of view, considering that the synthesized control law can be implemented satisfactorily in standard flight control systems. Conclusions with remarks towards the practical use of the method are presented as well.

1. Introduction

An interesting technique that has allowed linear parameter varying (LPV) control synthesis algorithms to obtain less conservative performance bounds is given the name of Full Block S-Procedure (FBSP). See [1, 2] and references therein. The formulas in [3, 4], render synthesis conditions combining the techniques of [1, 5], to develop a generalized framework for LPV control, based upon parameter dependent Lyapunov functions (PDLFs) and full block multipliers (FBMs). Yet these results provide linear matrix inequality (LMI) constrains with an infinite number of inequalities. In practice, one must resort to gridding the parameter variation set in order to apply the methods.

Subsequent work presented in [6] makes focus in particular form of PDLFs, namely, PDFLs that depend on the parameter in a linear fractional fashion, for systems whose open loop state space matrices depend on the parameter in a linear fractional way as well. In the sequel we will call the former LFT PDLFs and the latter LFT systems. From the practical point of view, the technique, which is based upon [4], is most appealing considering renders a set of constraints with a finite number of LMIs.

Another reference that is relevant from the practical point of view in LPV control is given by [7]. This work compares the results of [5] with an extension to LPV systems of the results in [8]. Helpful hints are given, in order to obtain LPV controllers that can be implemented in practice. The synthesis methods seek to establish stability and performance making use of PDLFs.

It must be recalled that synthesis conditions like the ones presented in [5] require a first step in order to establish feasibility of the problems and a second step to calculate the controllers. Under certain rank restrictions imposed on the state space matrices of the augmented plant, closed form expressions can be given for the calculation of the controllers. As opposed to this case, in [8] like results, an optimization is directly carried out, on a set of variables which are equivalent to the final state space matrices of the controller, through a nonlinear change of variables, which is the approach followed in this work.

It is rather frequent, in the application of LPV methods, that feasible problems with acceptable 𝛾-performance indexes (see [5, 9] for a definition) show undesirable transient response. With a slight abuse of terminology, it can be posed as LPV systems having closed loop “poles” in undesirable locations. It is actually the poles of each linear time invariant system (LTI) resulting from holding the parameter vector constant, that turn out to have a nonconvenient loci. It has been reported previously that LPV synthesis, in particular when approached through a single quadratic Lyapunov function (SQLF), shows a problem called “fast poles” (see [10]).

The aspect of transient response of systems has not been dealt with in the FBSP framework of [6]. The work presented here extends the application of the FBSP, together with LFT PDLFs, to the same kind of LFT systems [6] deals with. The proposed approach includes the possibility of having the closed loop poles of each LTI system resulting from LPV dynamics with constant parameter trajectories, to have their loci in a prescribed region. This idea is most appealing from the practical viewpoint, in order to carry out the simulation and implementation of controllers. An application to the problem of 6 degrees of freedom (DOF) control of a spacecraft is presented with numerical hardware in the loop (HIL) simulations, as an application example.

The paper is organized as follows. Section 2 presents the developed formulas for LPV control with closed loop regional pole placement and FBMs. In Section 3, the synthesis method is employed to design the 6 DOF control for a rocket. Concluding remarks are given in Section 4.

2. Synthesis Method

In this section the synthesis method of LPV control with FBMs, PDLFs, and regional pole placement constraints is presented.

2.1. Background

The set 𝒫⊂ℝ^𝑠 is such that for each 𝜃=(𝜃₁,…,𝜃_𝑠)∈𝒫, |𝜃_𝑖|≤1. On the other hand, for some 𝜈=(𝜈₁,…,𝜈_𝑠)∈ℝ^𝑠 with 𝜈_𝑖>0, all 𝜈=(𝜈₁,…,𝜈_𝑠)∈𝒱⊂ℝ^𝑠 are such that |𝜈_𝑖|≤𝜈_𝑖. An 𝑟_𝛼=(𝑟_𝛼1,…,𝑟_𝛼𝑠)∈ℕ^𝑠 defines the sets Θ_𝛼={Θ_𝛼=diag{𝜃₁𝐼_𝑟𝛼1,…,𝜃_𝑠𝐼_𝑟𝛼𝑠}∶𝜃∈𝒫} and Θ̇‌_𝛼={Θ̇‌_𝛼=diag{𝜃̇‌₁𝐼_𝑟𝛼1,…,𝜃̇‌_𝑠𝐼_𝑟𝛼𝑠}∶𝜃̇‌∈𝒱}. The number 𝑛_𝛼∶=∑^𝑠_𝑖=1𝑟_𝛼𝑖 is used later. In the sequel, any 𝑟_𝛽=(𝑟_𝛽1,…,𝑟_𝛽𝑠)∈ℕ^𝑠 will be regarded as defining a number 𝑛_𝛽 and a couple of sets Θ_𝛽, and Θ̇‌_𝛽, in the same fashion as 𝑛_𝛼, Θ_𝛼, and Θ̇‌_𝛼 before. With the subindex omitted, it will be just 𝑟=(𝑟₁,…,𝑟_𝑠)∈ℕ^𝑠, defining 𝑛_𝑝∶=∑^𝑠_𝑖=1𝑟_𝑖, Θ and Θ̇‌. Throughout this paper, systems state space matrices depend on 𝑠-dimensional parameter trajectories evolving in the set ℱ^𝜈_𝒫={𝜃∈𝒞¹(ℝ+,ℝ^𝑠)∶𝜃(𝑡)∈𝒫,𝜃̇‌(𝑡)∈𝒱,forall𝑡∈ℝ₊}.

The following lemma is crucial in order to use FBMs for LPV control.

Lemma 1 (Full Block S-Procedure). Let 𝒢(Θ)=Θ⋆𝐺₁₁𝐺₁₂𝐺₂₁𝐺₂₂∶=𝐺₂₂+𝐺₂₁Θ𝐼−𝐺₁₁Θ⁻¹𝐺₁₂(1) be a linear fractional transformation (LFT) where 𝐺₁₁, 𝐺₁₂, 𝐺₂₁ and 𝐺₂₂ are real matrices of compatible dimensions. Given a real symmetric matrix 𝑀, the quadratic matrix inequality 𝒢^𝑇(Θ)𝑀𝒢(Θ)<0(2) holds for all Θ∈Θ, if and only if there exists a real symmetric full-block multiplier Π such that for any Θ∈Θ, ∗∗^𝑇diag{Π,𝑀}⎡⎢⎢⎣𝐺₁₁𝐺₁₂𝐼0𝐺₂₁𝐺₂₂⎤⎥⎥⎦<0,𝐼Θ^𝑇Π𝐼Θ≥0.(3)

Proof. See [6].

Remark 1. Condition (3) consists of an infinite number of constraints. Considering Θ a compact set defined by its 2^𝑠 vertices, additional constraints can be added in order to turn (3) into a condition with a finite number of constraints. Namely, partition the multiplier Π as Π=Π₁₁Π₁₂Π^𝑇₁₂Π₂₂,(4) and request Π₂₂<0. Then condition (3) will be convex with respect to Θ. As a consequence, if for all vertices Θ_𝑖 of Θ, the following LMI constraints 𝐼Θ_𝑖^𝑇Π₁₁Π₁₂Π^𝑇₁₂Π₂₂𝐼Θ_𝑖≥0(5) are satisfied, then inequality (3) will be so itself. This remark is important from the computational point of view. Notice that, while acceptable in practice, the approach can be conservative. Moreover, as observed in [6], fulfillment of constraint (3) is achieved if it is further enforced that Π₁₁=−Π₂₂>0, with Π₁₁, Π₂₂ being symmetric, Π₁₂ being skew-symmetric, and all of them being commutable with all Θ in Θ.

In order to state the control problem, consider an LFT parameter-dependent plant:⎡⎢⎢⎣𝑥̇(𝑡)𝑒(𝑡)𝑦(𝑡)⎤⎥⎥⎦=⎡⎢⎢⎣𝒜(Θ(𝑡))ℬ₁(Θ(𝑡))ℬ₂(Θ(𝑡))𝒞₁(Θ(𝑡))𝒟₁₁(Θ(𝑡))𝒟₁₂(Θ(𝑡))𝒞₂(Θ(𝑡))𝒟₂₁(Θ(𝑡))𝒟₂₂(Θ(𝑡))⎤⎥⎥_{}⎦⎡⎢⎢⎣𝑥(𝑡)𝑑(𝑡)𝑢(𝑡)⎤⎥⎥⎦,(6) where Θ(𝑡)∈Θ, 𝑥̇, 𝑥∈ℝ^𝑛, 𝑑∈ℝ^𝑛𝑑 is the disturbance, 𝑒∈ℝ^𝑛𝑒 is the controlled output, 𝑢∈ℝ^𝑛𝑢 is the control input and 𝑦∈ℝ^𝑛𝑦 is the measurement for control. The underbraced state space matrices of (6) depend on the parameter 𝜃 in a linear fractional way as follows:⎡⎢⎢⎣𝐴𝐵₁𝐵₂𝐶₁𝐷₁₁𝐷₁₂𝐶₂𝐷₂₁𝐷₂₂⎤⎥⎥⎦+⎡⎢⎢⎣𝐵₀𝐷₁₀𝐷₂₀⎤⎥⎥⎦Θ(𝑡)𝐼−𝐷₀₀Θ(𝑡)⁻¹𝐶₀𝐷₀₁𝐷₀₂.(7) It is assumed that the LFT representation is well-posed; that is, (𝐼−𝐷₀₀Θ(𝑡)) is invertible for any allowable parameter values. It is also assumed that the triple (𝒜,ℬ₂,𝒞₂) is parameter-dependent stabilizable and detectable for all 𝜃∈ℱ^𝜈_𝒫. This guarantees the existence of a stabilizing output feedback LPV controller. The class of LPV controllers we are interested in is of the form𝑥̇_𝑘(𝑡)𝑢(𝑡)=⎡⎢⎣𝒜_𝑘Θ(𝑡),Θ̇‌(𝑡)ℬ_𝑘Θ(𝑡),Θ̇‌(𝑡)𝒞_𝑘Θ(𝑡),Θ̇‌(𝑡)𝒟_𝑘Θ(𝑡),Θ̇‌(𝑡)⎤⎥⎦𝑥_𝑘(𝑡)𝑦(𝑡),(8) where 𝑥_𝑘∈ℝ^𝑛𝑘. The dimension of controller state 𝑛_𝑘 is yet to be determined.

The synthesis method used in this paper is based upon the results in [6] for LPV systems and [8] for LTI ℋ_∞ synthesis with pole clustering. The following definition is taken from [8].

Definition 1 (LMI-region). A subset 𝒟 of the complex plane is called an LMI region if there exist a symmetric matrix 𝛼=[𝛼_𝑘𝑙]∈ℝ^𝑚×𝑚 and a matrix 𝛽=[𝛽_𝑘𝑙]∈ℝ^𝑚×𝑚 such that 𝒟={𝑧∈ℂ∶𝑓_𝒟(𝑧)<0} with 𝑓_𝒟(𝑧)=𝛼+𝑧𝛽+𝑧𝛽^𝑇=𝛼_𝑘𝑙+𝛽_𝑘𝑙𝑧+𝛽_𝑙𝑘𝑧_{1≤𝑘,𝑙≤𝑚}.(9)

These regions make up a dense subset in the set of regions of the complex plane, symmetric with respect to the real axis. This makes them appealing for specifying pole placement design objectives.

Theorem 1 (LPV basic characterization with pole placement constraints). Let 𝑟_𝒮=(𝑟^𝒮₁,…,𝑟^𝒮_𝑠) and 𝑟_ℛ=(𝑟^ℛ₁,…,𝑟^ℛ_𝑠)∈ℕ^𝑠 define 𝑛_𝒮, 𝑛_ℛ, Θ_𝒮, Θ̇‌_𝒮 and Θ_ℛ, Θ̇‌_ℛ as in the beginning of this section. Let 𝒮(Θ_𝒮)=𝒯_𝒮(Θ_𝒮)^𝑇𝑄𝒯_𝒮(Θa_𝒮) and ℛ(Θ_ℛ)=𝒯_ℛ(Θ_ℛ)^𝑇𝑃𝒯_ℛ(Θ_ℛ), with 𝒯_𝒮Θ_𝒮=Θ_𝒮⋆𝑇_𝒮11𝑇_𝒮12𝑇_𝒮21𝑇_𝒮22,𝒯_ℛΘ_ℛ=Θ_ℛ⋆𝑇_ℛ11𝑇_ℛ12𝑇_ℛ21𝑇_ℛ22,(10) be two symmetric and positive definite matrix functions, where constant matrices 𝑄 and 𝑃 are yet to be found, with Θ_𝒮∈Θ_𝒮 and Θ_ℛ∈Θ_ℛ. Consider the LPV plant governed by (6), with parameter trajectories in ℱ^𝜈_𝒫. Suppose that there exist parameter dependent symmetric matrices 𝒮 and ℛ such as (10), a positive real number 𝛾, and a parameter dependent quadruple of state space data 𝒜_𝑘, ℬ_𝑘, 𝒞_𝑘, and 𝒟_𝑘, such that the following LMI constraints −ℛ𝐼𝐼𝒮<0,(11)𝛼_𝑘𝑙ℛ𝐼𝐼𝒮+𝛽_𝑘𝑙Φ+𝛽_𝑙𝑘Φ^𝑇_𝑘,𝑙<0,(12)Ψ=Ψ₁₁Ψ^𝑇₂₁Ψ₂₁Ψ₂₂<0(13) are satisfied for all 𝜃∈𝒫, 𝜃̇‌∈𝒱, with Φ=⎡⎢⎣𝒜ℛ+ℬ₂𝒞_𝑘𝒜+ℬ₂𝒟_𝑘𝒞₂𝒜_𝑘𝒮𝒜+ℬ_𝑘𝒞₂⎤⎥⎦,Ψ₁₁=⎡⎢⎢⎣−ℛ̇‌+𝒜ℛ+ℛ𝒜^𝑇+ℬ₂𝒞_𝑘+ℬ₂𝒞_𝑘^𝑇ℬ₁+ℬ₂𝒟_𝑘𝒟₂₁ℬ₁+ℬ₂𝒟_𝑘𝒟₂₁^𝑇−𝛾𝐼⎤⎥⎥⎦,Ψ₂₁=⎡⎢⎣𝒜_𝑘+𝒜+ℬ₂𝒟_𝑘𝒞₂^𝑇𝒮ℬ₁+ℬ_𝑘𝒟₂₁𝒞₁ℛ+𝒟₁₂𝒞_𝑘𝒟₁₁+𝒟₁₂𝒟_𝑘𝒟₂₁⎤⎥⎦,Ψ₂₂=⎡⎢⎣𝒮̇‌+𝒮𝒜+𝒜^𝑇𝒮+ℬ_𝑘𝒞₂+ℬ_𝑘𝒞₂^𝑇𝒞₁+𝒟₁₂𝒟_𝑘𝒞₂^𝑇𝒞₁+𝒟₁₂𝒟_𝑘𝒞₂−𝛾𝐼⎤⎥⎦.(14) Then, there exists a gain-scheduled output-feedback controller as (8) such that one has the following. (1)Internal stability is enforced.(2)𝛾 is a bound on the 𝐿₂ gain of the closed-loop system given by the interconnection of (6) with (8).(3)The poles of each closed-loop LTI system, resulting from all constant parameter trajectories in ℱ^𝜈_𝒫, are circumscribed to an LMI region prescribed by a characteristic function such as (9).

Proof. See [8, 11].

For the optimization problem to be convex, this method seeks a unique closed loop Lyapunov Matrix 𝒳 simultaneously valid for 𝐿₂-gain and pole placement conditions. 𝒳 can be computed from 𝒮 and ℛ. As mentioned in [8], this approach is potentially conservative, but rarely in practice.

2.2. Main Results

Next, to proceed towards the derivation of synthesis conditions, a dependence of the so-called transformed controller matrices 𝒜_𝑘, ℬ_𝑘, 𝒞_𝑘, and 𝒟_𝑘 on the measured parameter vector (𝜃) is proposed. Let 𝑟_𝑎𝑘=(𝑟^𝑎𝑘₁,…,𝑟^𝑎𝑘_𝑠) and 𝑟_𝑐𝑘=(𝑟^𝑐𝑘₁,…,𝑟^𝑐𝑘_𝑠)∈ℕ^𝑠 define the numbers 𝑛_𝑎𝑘 and 𝑛_𝑐𝑘, and the sets Θ_𝑎𝑘, Θ̇‌_𝑎𝑘 and Θ_𝑐𝑘, Θ̇‌_𝑐𝑘 as in the beginning of this section. The transformed controller matrices will be given by𝒜_𝑘=𝒯_𝑎𝑘𝐴_𝑘,ℬ_𝑘=𝒯_𝑎𝑘𝐵_𝑘,𝒞_𝑘=𝒯_𝑐𝑘𝐶_𝑘,𝒟_𝑘=𝒯_𝑐𝑘𝐷_𝑘(15)

with 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, and 𝐷_𝑘 being constant matrices and𝒯_𝑎𝑘Θ_𝑎𝑘=Θ_𝑎𝑘⋆𝑇_𝑎𝑘11𝑇_𝒮12𝑇_𝒮21𝑇_𝒮22,Θ_𝑎𝑘∈𝚯_𝑎𝑘,𝒯_𝑐𝑘Θ_𝑐𝑘=Θ_𝑐𝑘⋆𝑇_ℛ11𝑇_ℛ12𝑇_ℛ21𝑇_ℛ22,Θ_𝑐𝑘∈𝚯_𝑐𝑘.(16)

In order to apply the FBSP on LMIs (11), (12), and (13) in a way resembling theorem 4 in [6], the following lemma is presented.

Lemma 2. Let Θ_ℛ∈Θ_ℛ, Θ_𝒮∈Θ_𝒮, Θ̇‌_𝒮∈Θ̇‌_𝒮, Θ̇‌_ℛ∈Θ̇‌_ℛ, Θ_𝑎𝑘∈Θ_𝑎𝑘, Θ_𝑐𝑘∈Θ_𝑐𝑘, and Θ∈Θ. LMIs (11), (12) and (13) can be rewritten as 𝒢^𝑇(−𝑀)𝒢<0,𝒢^𝑇_𝑝𝑝𝑀_𝑝𝑝𝒢_𝑝𝑝<0,𝒢^𝑇_∞𝑀_∞𝒢_∞<0.(17) The 𝒢, 𝒢_𝑝𝑝, and 𝒢_∞ are LFTs depending on the open loop data (6) and on the 𝒯_𝒮, 𝒯_ℛ, 𝒯_𝑎𝑘, and 𝒯_𝑐𝑘 functions. They can be expressed as 𝒢=Θ⋆𝐺₁₁𝐺₁₂𝐺₂₁𝐺₂₂,𝒢_𝑝𝑝=Θ_𝑝𝑝⋆𝐺_𝑝𝑝11𝐺_𝑝𝑝12𝐺_𝑝𝑝21𝐺_𝑝𝑝22,𝒢_∞=Θ_∞⋆𝐺_∞11𝐺_∞12𝐺_∞21𝐺_∞22(18) with Θ=diag(Θ_ℛ,Θ_𝒮), Θ_𝑝𝑝=diag(Θ_𝒮,Θ_ℛ,Θ_𝑎𝑘,Θ_𝑐𝑘,Θ) and Θ_∞=diag(Θ̇‌_𝒮,Θ̇‌_ℛ,Θ_𝒮,Θ_ℛ,Θ_𝑎𝑘,Θ_𝑐𝑘,Θ). On the other hand, the 𝑀, 𝑀_𝑝𝑝, and 𝑀_∞ matrices depend on the 𝛼 and 𝛽𝑚×𝑚 matrices that specify a design LMI region, as in Definition 1, and on (1)a pair of symmetric positive definite 𝑃 and 𝑄 matrices of the 𝒮 and ℛ matrix functions,(2)a quadruple 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, and 𝐷_𝑘 of controller state space data,(3) a real positive performance index𝛾,where the enumerated objects are to be determined in the optimization process.

Proof. See [12] for the definition of matrices 𝑀, 𝑀_𝑝𝑝, 𝑀_∞, 𝒢, 𝒢_𝑝𝑝, and 𝒢_∞ and for the proof, which can be obtained through tedious but straightforward matrix calculations based upon the results in [6, 8].

Theorem 2 (LPV control with pole placement constraints and FBMs). The inequalities of (17) are satisfied, if and only if there exist symmetric positive definite real matrices 𝑃 and 𝑄, a performance index 𝛾, a quadruple of controller matrices 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, and 𝐷_𝑘, and symmetric full block multipliers Π, Π_𝑝𝑝 and Π_∞ such that the following LMIs ∗^𝑇diag{Π,𝑀}𝐺<0,∗^𝑇diagΠ_𝑝𝑝,𝑀_𝑝𝑝𝐺_𝑝𝑝<0,∗^𝑇diagΠ_∞,𝑀_∞𝐺_∞<0,𝐺=⎡⎢⎢⎣𝐺₁₁𝐺₁₂𝐼0𝐺₂₁𝐺₂₂⎤⎥⎥⎦,𝐺_𝑝𝑝=⎡⎢⎢⎣𝐺_𝑝𝑝11𝐺_𝑝𝑝12𝐼0𝐺_𝑝𝑝21𝐺_𝑝𝑝22⎤⎥⎥⎦,𝐺_∞=⎡⎢⎢⎣𝐺_∞11𝐺_∞12𝐼0𝐺_∞21𝐺_∞22⎤⎥⎥⎦(19) are satisfied, and for all 𝜃∈𝒫 and 𝜃̇‌∈𝒱, the following conditions 𝐼Θ_𝑃^𝑇Π_𝑃𝐼Θ_𝑃≥0,⎡⎢⎣𝐼Θ_𝑄⎤⎥⎦^𝑇Π_𝑄⎡⎢⎣𝐼Θ_𝑄⎤⎥⎦≥0,𝐼Θ^𝑇Π𝐼Θ≥0(20) are met.

Proof. The application of Lemma 1 to inequalities (17) produces the desired result.

The computation of the controller's state space matrices is carried out following the algorithm prescribed in [7]. All remarks made in that paper, concerning the use of PDLFs, which aim towards obtaining controllers whose state space matrices do not depend on the parameter rate of variation, are applicable here as well (see [7, Table I]).

As observed in [6], in practice, the considerations of Remark 1 can lead to significantly reducing the number of decision variables of the problem. Degradation of the computed performance bound 𝛾 is also a possibility.

3. Application Example

3.1. Nonlinear Model

The example we consider is a sounding rocket (see Figure 1) which should follow a prescribed trajectory. A dynamic model is presented (see Figure 2), describing the position, velocity, orientation, and angular velocity errors of the actual vehicle (frame 𝐁) with respect to the prescribed trajectory (frame 𝐃). The differential gravity force is neglected in this error model for practical reasons.

(a)

(b)

Figure 1 

Vehicle diagram.

Figure 2 

Position of frame 𝐁 with respect to frame 𝐃.

A simple model of aerodynamic drag and lift forces taken from [13] is included. According to this model, these forces depend on the dynamic pressure, the angles of attack 𝛼, and sideslip 𝛽 (see [14]). The moments resulting from the aerodynamic forces are computed under the assumption that the center of pressure (CP) is located above the CM. This renders unstable aerodynamics posing a challenge on the control system (see Figure 1). 𝐹^𝑥_𝑎, 𝐹^𝑦_𝑎, and 𝐹^𝑧_𝑎 denote the aerodynamic forces acting on the CP in frame 𝐁.

The actuator considered for the rocket is a nozzle gimbal which allows small rotations around the 𝑧-and 𝑦-axes. A couple of gas jets are placed to generate torques around the 𝑥-axis. Hence, the actual control inputs of the plant will be the thrust, the rotation angles of the gimbal, and the torque exerted by the jets. For small rotations of the gimbal, a change of variables is carried out, and as a consequence, the control inputs of the plant model used in the controller design will be denoted by 𝐹^𝑥_𝑡ℎ, 𝐹^𝑦_𝑡ℎ, 𝐹^𝑧_𝑡ℎ, and 𝑚_𝑥 assuming that the forces are applied at the tail (hence providing torques in the 𝑦 and 𝑧 axes). The following terms will be used to denote the actuator and aerodynamic forces and moments in the state space equations: 𝐹=(𝐹^𝑥_𝑡ℎ+𝐹^𝑥_𝑎)(𝐹^𝑦_𝑡ℎ+𝐹^𝑦_𝑎)(𝐹^𝑧_𝑡ℎ+𝐹^𝑧_𝑎)^𝑇,𝑀=𝑚_𝑥−(ℓ𝐹^𝑧_𝑡ℎ+ℓ₁𝐹^𝑧_𝑎)(ℓ𝐹^𝑦_𝑡ℎ+ℓ₁𝐹^𝑦_𝑎)^𝑇.(21)

𝐹_𝑑 and 𝑀_𝑑 denote the counterparts of 𝐹 and 𝑀 for the desired trajectory. As mass variation concerns, it is assumed that the thrust force acting on the vehicle comes from fuel exhaustion and that the inertia matrix varies uniformly with it, that is, the CM does not change. 𝐽=diag[𝐽_𝑥𝐽_𝑡𝐽_𝑡] denotes the diagonal varying inertia matrix. The variables for the vehicle dynamics will be 𝑟, 𝑣, [𝑞₀,𝑞], and 𝜔 which, respectively, give the position, velocity, orientation and angular velocity of frame 𝐁 relative to frame 𝐃. Under these assumptions, the error dynamics of the rocket are𝑟̇=𝑣,𝑣̇‌=1𝑚^𝑑𝐶_𝑏𝐹−𝐹_𝑑,𝑞̇=12𝑞₀𝜔+𝑞×𝜔,𝜔̇=𝐽⁻¹^𝑑𝐶_𝑏𝑀−𝑀_𝑑,𝑞₀̇=−12𝑞⋅𝜔(22) with ^𝑑𝐶_𝑏 denoting the rotation matrix from frame 𝐁 to 𝐃.

3.2. Linearization: LPV Model

Linearization of (22) will be carried out, in order to obtain an LPV model of the plant, under a few extra assumptions related to nominal control and aerodynamic forces. During ascent, the vehicle should nominally follow a zero angle of attack trajectory, hence only withstanding drag. It can be shown that the nominal actuator forces acting on the 𝑦-and 𝑧-axes of the vehicle are negligible as compared to the actual forces that compensate for disturbances when the actual angle of attack deviates from zero. It can also be accepted that 𝑚_𝑥 is nominally zero. Under these assumptions 𝑀_𝑑=0, and 𝐹_𝑑=[(𝑓_𝑁−𝐷)00]^𝑇 where 𝑓_𝑁 is the nominal thrust and 𝐷 is the drag force. Let 𝑢≜[𝐹^𝑥_𝑡ℎ𝐹^𝑦_𝑡ℎ𝐹^𝑧_𝑡ℎ𝑚_𝑥]^𝑇 and 𝑢_𝑑≜[𝑓_𝑁000]^𝑇. Let 𝑟_𝑖, 𝑣_𝑖, 𝑞_𝑖, and 𝜔_𝑖 (𝑖=1,2,3) be the components of the corresponding vectors, and the state variable is 𝑥≜[𝑥^𝑇₁𝑥^𝑇₂𝑥^𝑇₃𝑥^𝑇₄]^𝑇 with 𝑥₁=[𝑟₁𝑣₁]^𝑇, 𝑥₂=[𝑟₂𝑣₂𝑞₃𝜔₃]^𝑇, 𝑥₃=[𝑟₃𝑣₃𝑞₂𝜔₂]^𝑇, and 𝑥₄=[𝑞₁𝜔₁]^𝑇. The differential control is 𝛿_𝑢=𝑢−𝑢_𝑑. With these definitions in mind we seek to linearize (22) which can be rewritten as𝑥̇=𝑓𝑥,𝛿_𝑢,𝜃=𝐴(𝜃)𝑥+𝐵(𝜃)𝛿_𝑢+𝑜𝑥,𝛿_𝑢,𝜃,(23) where 𝑜(𝑥,𝛿_𝑢,𝜃) represents the higher-order terms of the series expansion of 𝑓 around (0,0,𝜃). 𝐴(𝜃) and 𝐵(𝜃) are the following Jacobians: 𝐴(𝜃)≜𝜕𝑓𝑥,𝛿_𝑢,𝜃𝜕𝑥|||||_(0,0,𝜃)𝐵(𝜃)≜𝜕𝑓(𝑥,𝛿_𝑢,𝜃)𝜕𝛿_𝑢||||_(0,0,𝜃).(24) The LPV parameter 𝜃 is a function of mass, dynamic pressure, and the state variables of the actual vehicle. Mass variations can be estimated with a model of fuel consumption, and dynamic pressure can be estimated from inertial position and velocity measurements as well as from a model of the atmosphere. Hence, 𝜃 can be known in real time as if it were measured.

An evaluation of the Jacobian matrices of (24) through a symbolic manipulation software package shows that 𝐴(𝜃) and 𝐵(𝜃) are affine in the parameters, as follows:𝜃₁≜2𝑄₀𝑚𝑆,𝜃₂≜1𝑚.(25)

It also shows that the system can be decomposed in four decoupled systems corresponding to the 𝑥_𝑖 parts of the state variable 𝑥 (𝑖=1,2,3,4). As a consequence, the control problem can be split in four 𝛾-performance problems with pole placement constraints with each subsystem having a state space representation as follows (𝑖=1,2,3,4):𝑥̇_𝑖=𝐴_𝑖(𝜃)𝑥+𝐵_𝑖(𝜃)𝛿_𝑢𝑖,𝜃=𝜃₁𝜃₂^𝑇.(26)𝜃 evolves in the parameter variation set which is a square in ℝ². Mass and dynamic pressure variations throughout the nominal trajectory determine the bounds for 𝜃₁ and 𝜃₂.

The aerodynamic forces are included in the error dynamics in a first-order approximation for zero angle of attack nominal trajectories as follows:𝐹^𝑧_𝑎=⎛⎜⎜⎝𝜕𝐹^𝑧_𝑎𝜕𝑞_𝑦||||_(0,0,𝜃)⎞⎟⎟⎠𝑞_𝑦,𝐹^𝑦_𝑎=⎛⎜⎜⎝𝜕𝐹^𝑦_𝑎𝜕𝑞_𝑧||||_(0,0,𝜃)⎞⎟⎟⎠𝑞_𝑧.(27)

These terms end up inside the 𝐴_𝑖 matrices of the LPV plants for 𝑖=2,3 with the derivatives depending on 𝜃₁. Since drag has a maximum for the nominal trajectory, its variations are neglected in the first-order approximation. Hence, aerodynamics only affect subsystems 2 and 3.

3.3. Controller Synthesis

As a consequence of decoupling, the control problem is reduced to synthesizing one controller per subsystem. This simplifies the statement since systems 𝐺₁ and 𝐺₄ depend only on parameter 𝜃₂.

For each subsystem an augmented plant as the one in (6) must be specified. In this problem, for all subsystems, the disturbance signal is split into a part which represents disturbance forces or moments, and a part which represents measurement noise. The performance signal is divided in two parts as well. One of them involves the state variables and it represents the design's commitment with disturbance rejection at the output. The other one involves the control force and is included in order to achieve disturbance rejection with reasonable control action.

For each subsystem the augmented plant is the result of the block interconnection of Figure 3. The state space representation of each 𝐺_𝑖 block is as follows:𝑥̇_𝑖=𝒜_𝑖(𝜃)𝑥_𝑖+ℬ_𝑑𝑖𝜃₂𝑑₁+ℬ_𝑖𝜃₂𝛿_𝑢𝑖,𝑦_𝑖̃=𝑥_𝑖.(28)

Figure 3 

Block interconnection making up the augmented plant (left) and Inclusion of low-pass transfer function model of actuator dynamics (right).

For the sake of clarity as notation regards, note that the ℬ_𝑖 matrices of the systems described by (28) play the role of matrix ℬ₂ in the system described by (6). For the final augmented plants (𝑃_𝑖, Figure 3) the lowpass transfer function 𝑇_act(𝑠)=1/((𝑠/𝜔_𝑎)+1), with 𝜔_𝑎=2𝜋⋅10rad/sec, is included in order to model actuator dynamics. As the 𝐵_𝑑𝑖 matrix concerns, it is assumed for all systems that 𝐵_𝑑𝑖=𝜉_𝑖𝐵_𝑖 with 𝜉_𝑖 being a design parameter (1<𝜉_𝑖<10). The 𝜉_𝑖 gives the ratio between the disturbance and control forces or torques.

System 𝐺₁ involves longitudinal dynamics. The matrices that make up the plant are as follows:𝐴₁=0100,𝑊_𝑤=5000.2,𝐵₁=0𝜃₂(29)

with 𝑊₁=𝑊⁻¹_𝑤, 𝑘_𝑢1=10⁻² and 𝜉₁=10.

Systems 𝐺₂ and 𝐺₃ involve lateral dynamics, with their matrices being𝐴₂=⎡⎢⎢⎢⎢⎣010000−𝐷_𝛼𝜃₁+2𝑓_𝑁𝜃₂0000120000⎤⎥⎥⎥⎥⎦,𝐴₃=⎡⎢⎢⎢⎢⎣010000𝐷_𝛼𝜃₁−2𝑓_𝑁𝜃₂0000120000⎤⎥⎥⎥⎥⎦,𝐵₂=⎡⎢⎢⎢⎢⎢⎣0𝜃₂0−𝑚₀𝐼_𝑡0𝜃₂⎤⎥⎥⎥⎥⎥⎦,𝐵₃=⎡⎢⎢⎢⎢⎢⎣0𝜃₂0𝑚₀𝐼_𝑡0𝜃₂⎤⎥⎥⎥⎥⎥⎦,𝑊_𝑤=⎡⎢⎢⎢⎢⎢⎣500000.20000𝜋1800000.1𝜋180⎤⎥⎥⎥⎥⎥⎦(30)

with 𝑊₁=𝑊⁻¹_𝑤, 𝑘_𝑢2=𝑘_𝑢3=10⁻², and 𝜉₂=𝜉₃=10.

System 𝐺₄ involves roll dynamics. The matrices that make up the plant are as follows:𝐴₄=⎡⎢⎣01200⎤⎥⎦,𝑊_𝑤=⎡⎢⎢⎣𝜃18000.1𝜃180⎤⎥⎥⎦,𝐵₄=⎡⎢⎢⎣0𝑚₀𝐼_𝑥0𝜃₂⎤⎥⎥⎦(31) with 𝑊₁=𝑊⁻¹_𝑤, 𝑘_𝑢4=10⁻², and 𝜉₄=10.

3.4. Synthesis Results

The synthesis procedure was carried out on all subsystems. The pole placement region used was 𝒟={𝑧∈ℂ,−2𝜋⋅8rad/sec<ℜ(𝑧)}, which takes into account practical aspects such as ease of implementation and simulation. The vehicle's details can be seen in Table 1. In this example, focus is made on demonstrating the validity of the presented synthesis method. As a consequence, in order to compare the results obtained here with respect to previous work [10], SQLFs were employed on each subsystem (i.e., 𝒮 and ℛ constant). Another design decision that was taken concerns the choice of the 𝒯_𝑎𝑘 and 𝒯_𝑐𝑘 functions which prescribe the way controller matrices dependend on the parameter. For the example these functions were picked as 𝒯_𝑎𝑘=𝐼_𝑛𝜃₁𝐼_𝑛𝜃₂𝐼_𝑛,𝒯_𝑐𝑘=𝐼_𝑛𝑢𝜃₁𝐼_𝑛𝑢𝜃₂𝐼_𝑛𝑢.(32)

Table 1 

Vehicle's details.

This choice was made, in order to have the same kind of affine parameter dependence of the original plant in the controller. Once the controllers for each subsystem were synthesized, they were appended to make up the complete LPV controller.

A heuristic approach to address the “fast poles” problem, while using algorithms as the one presented in [6], is to bound the trace of the ℛ and 𝒮 matrices. A bound in the range of the decision variables could be imposed alternatively. The method is occasionally successful but offers no guarantee. In this particular application, this heuristic was useless in order to prevent one single “fast pole” in the controller, per subsystem. Another heuristic was tried, consisting in the residualization of the “fast pole” in the LTI part of the controller. Nevertheless, this approach showed poor closed loop transient behavior in non-real-time simulations. As a consequence, it was dropped. On the other hand, the controller synthesized using the proposed method showed better time responses with lower overshoot. The performance 𝛾s with the standard algorithm [6] were in the order of the ones obtained with the proposed one. The overall conclusion is that the method proposed enhances the capabilities of the standard LPV technique for real-world applications.

A considerable increase in the number of decision variables can take place. For this application, with the pole placement region being a half plane, the number of decision variables of the optimization problem for each lateral controller goes from 73 with the algorithm of [6] to 1363 in this case. A more careful and potentially conservative restriction of the number of multiplier variables, in line with Remark 1, could reduce this number. This was not done in this case, given that the computation time using standard PC hardware was acceptable and considering that the number of decision variables does not influence the implementation of the controller.

With respect to previous work [10], the improvement in the performance bounds is a remarkable result (𝛾=1.89 versus 162 for subsystem 1, 𝛾=30.46 versus 96 for subsystems 2 and 3, and 𝛾=1.29 versus 169 for subsystem 4). Moreover, an improvement in the reduction of the sampling frequency from 125 to 45.45 Hz due to the feasible relocation of closed loop poles was achieved.

3.5. HIL Simulations

To stress the fact that the method is not only valid but also applicable, real-time numerical simulations were carried out. The control law was implemented in a computer based upon an Atmel TSC695E SPARC7 class microprocessor operating at 20 MHz. This 32-bit microcontroller has been available in commercial space systems for more than a decade, setting a de facto standard. The setup for simulation is fairly simple, consisting of an Atmel VAB695E Evaluation Kit, with an add-on board containing an Atmel AT7908E CANbus (ISO 11898-1) controller, connecting this computer with a standard PC featuring a Peak-PCAN pci CANbus interface board. The use of CANbus in aerospace applications is fairly standard [15]. The simulation of the vehicle's dynamics is carried out on the PC using GNU Scientific Library [16], for the real-time simulation of the vehicle’s dynamics. The PC features Xenomai Linux [17] as Real-Time Operating System. GNU Scientific Library is employed in the VAB695E software as well, in single precision floating point mode, for the implementation of the LPV control law. RTEMS real-time operating system [18] features the VAB695E board.

To evaluate the response of the system to disturbance signals the 𝐿₂-gain criterion accounts for, the PC simulates sensor measurements corrupted by colored, weighted, pseudorandom noise. An input disturbance is introduced as well consisting of colored, weighted, pseudorandom lateral forces. These corrupted simulated measurements are transmitted upon request from the control computer. The VAB695E computes control inputs in real time, transmitting commands to actuators simulated in the PC. This renders simulated real time closed loop operation.

The simulations carried out show responses with the initial conditions deviated from the nominal ones as follows: 𝑥(0)=𝑟(0)^𝑇𝑣(0)^𝑇𝑞₀(0)𝑞(0)^𝑇𝜔(0)^𝑇^𝑇(33)

with 𝑟(0)=6378000𝟑𝟎𝟎𝟑𝟎𝟎^𝑇,𝑣(0)=0.𝟏.𝟏^𝑇,𝑞(0)=0.𝟎𝟑𝟏𝟔0^𝑇,𝜔(0)=000^𝑇,𝑞₀(0)=.𝟗𝟗𝟓.(34)

All quantities in bold face should nominally be zero, except for 𝑞₀ which would be nominally one. The prescribed trajectory is a vertical ascent from the north pole. Disturbance of the initial conditions is useful in order to evaluate transient behavior.

In Figure 4 the time history of the state variables can be seen. Figure 5 shows the control error forces and commands. The gimbal’s rotations (𝛿_𝑦 and 𝛿_𝑧) are simulated with a saturation at 5^∘. Note nevertheless that they remain unsaturated practically throughout the whole simulation. The discrete time implementation of the controller, which is synthesized in a continuous time framework, is carried out through a zero-order hold transformation on the LTI part of the LFT controller (see [19]). The sampling frequency picked for this was 45.45 Hz (22-millisecond sampling time).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 4 

Response to initial conditions. Error State Variables versus time (sec). (a), (b), and (c): position components (m). (d), (e), and (f): velocity components (m/sec)). (g), (h), and (i): quaternion components (deg). (j), (k), and (l): angular velocity components (rad/sec).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5 

Response to initial conditions. Column 1: error Control Forces 𝐹𝑥 (N), 𝐹𝑦 (N), 𝐹𝑧 (N), and 𝑚𝑥 (Nm) versus time (sec). Column 2: error commands 𝑓𝑒 (N), 𝛿𝑦 (deg), 𝛿𝑧 (deg) and 𝑚𝑥(Nm) versus time as well.

4. Conclusions

In this work, the use of FBMs was extended to LPV synthesis with regional pole placement constraints. The usefulness of the method was tested on an application example with HIL simulations. The design of an LPV controller for a 6 DOF vehicle with pole placement constraints shows an adequate response without degrading the 𝛾 performance index. The system’s LPV “poles” were satisfactorily placed acceptably increasing the computational cost of design.

Acknowledgments

This research was partially supported by the Universidad Nacional de Quilmes, Argentina, through Grant PUNQ 0530/07. The setup for simulation was assembled at the laboratories of CONAE, the Argentine Space Agency. The second author has been supported by CONICET and a PRH Grant from the Ministry of Science and Technology of Argentina. The authors wish to acknowledge the contribution of Dr. Ke Dong to this work, who gently sent the source code of the software developed for the example in [6], for us to see.