Model Predictive Control of Uncertain Constrained Linear System Based on Mixed ℋ₂/ℋ_∞ Control Approach

Abstract

Uncertain constrained discrete-time linear system is addressed using linear matrix inequality based optimization techniques. The constraints on the inputs and states are specified as quadratic constraints but are formulated to capture hyperplane constraints as well. The control action is of state feedback and satisfies the constraints. Uncertainty in the system is represented by unknown bounded disturbances and system perturbations in a linear fractional transform (LFT) representation. Mixed ℋ₂/ℋ_∞ method is applied in a model predictive control strategy. The control law takes account of disturbances and uncertainty naturally. The validity of this approach is illustrated with two examples.

1. Introduction

Model predictive control (MPC) is a class of model-based control theories that use linear or nonlinear process models to forecast system behaviour. MPC is one of the control techniques that is able to cope with model uncertainties in an explicit way [1]. MPC has been used widely in practical applications to industrial process systems [2] and active vibration control of railway vehicles [3]. One of the methods used in MPC when uncertainties are present is to minimise the objective function for the worst possible case. This strategy is known as minimax and was originally proposed [4] in the context of robust receding control, [5] in the context of feedback and feedforward control and [6] in the context of ℋ_∞ MPC. MPC has been applied to ℋ_∞ problems in order to combine the practical advantage of MPC with the robustness of the ℋ_∞ control, since robustness of MPC is still being investigated for it to be applied practically.

This work is motivated by the work in [7, 8] where uncertainty in the system was modeled by perturbations in a linear fractional representation. In [9], model predictive control based on a mixed ℋ₂/ℋ_∞ control approach was considered. The designed controller has the form of state feedback and was constructed from the solution of a set of feasibility linear matrix inequalities. However, the issue of handling both uncertainty and disturbances simultaneously was not considered. In this paper, we extend the result of [9] to constrained uncertain linear discrete-time invariant systems using a mixed ℋ₂/ℋ_∞ design approach and the uncertainty considered is norm-bounded additive. This is more suitable as both performance and robustness issues are handled within a unified framework.

The method presented in this paper develops an LMI design procedure for the state feedback gain matrix 𝐹, allowing input and state constraints to be included in a less conservative manner. A main contribution is the accomplishment of a prescribed disturbance attenuation in a systematic way by incorporating the well-known robustness guarantees through ℋ_∞ constraints into the MPC scheme. In addition, norm-bounded additive uncertainty is also incorporated. A preliminary version of some of the work presented in this paper was presented in [10].

The structure of the work is as follows. After defining the notation, we describe the system and give a statement of the mixed ℋ₂/ℋ_∞ problem in Section 2. In Section 3, we derive sufficient conditions, in the form of LMIs, for the existence of a state feedback control law that achieves the design specifications. In Section 4, we consider two examples that illustrate our algorithm. Finally, we conclude in Section 5.

The notation we use is fairly standard. ℛ denotes the set of real numbers, ℛ^𝑛 denotes the space of 𝑛-dimensional (column) vectors whose entries are in ℛ and ℛ^𝑛×𝑚 denoting the space of all 𝑛×𝑚 matrices whose entries are in ℛ. For 𝐴∈ℛ^𝑛×𝑚, we use the notation 𝐴^𝑇 to denote transpose. For 𝑥,𝑦∈ℛ^𝑛, 𝑥<𝑦 (and similarly ≤, >, and ≥) is interpreted element wise. The identity matrix is denoted as 𝐼 and the null matrix by 0 with the dimension inferred from the context.

2. Problem Formulation

We consider the following discrete-time linear time invariant system:𝑥_𝑘+1=𝐴𝑥_𝑘+𝐵_𝑤𝑤_𝑘+𝐵_𝑢𝑢_𝑘+𝐵_𝑝𝑝_𝑘,𝑞_𝑘=𝐶_𝑞𝑥_𝑘+𝐷_𝑞𝑢𝑢_𝑘+𝐷_𝑞𝑤𝑤_𝑘,𝑝_𝑘=Δ_𝑘𝑞_𝑘,𝑧_𝑘=⎡⎢⎢⎣𝐶_𝑧𝑥_𝑘𝐷_𝑧𝑢𝑢_𝑘⎤⎥⎥⎦,𝑥₀given,(1) where 𝑥₀ is the initial state, 𝑥_𝑘∈ℛ^𝑛 is the state, 𝑤_𝑘∈ℛ^𝑛𝑤 is the disturbance, 𝑢_𝑘∈ℛ^𝑛𝑢 is the control, 𝑧_𝑘∈ℛ^𝑛𝑧 is the controlled output, 𝐴∈ℛ^𝑛×𝑛, 𝐵_𝑤∈ℛ^{𝑛×𝑛𝑤}, 𝐵_𝑢∈ℛ^{𝑛×𝑛𝑢}, 𝐶_𝑧∈ℛ^{𝑛𝑧1×𝑛}, and 𝐷_𝑧𝑢∈ℛ^{𝑛𝑧2×𝑛𝑢}, and where 𝑛_𝑧=𝑛_𝑧1+𝑛_𝑧2. The signals 𝑞_𝑘 and 𝑝_𝑘 model uncertainties or perturbations appearing in the feedback loop.

The operator, Δ_𝑘, is block diagonal:Δ_𝑘∈𝚫_𝑘=⎧⎪⎪⎨⎪⎪⎩Δ_𝑘=⎡⎢⎢⎢⎢⎣Δ_1𝑘0⋱0Δ_𝑡𝑘⎤⎥⎥⎥⎥⎦∶‖‖Δ_𝑖𝑘‖‖≤1∀𝑖⎫⎪⎪⎬⎪⎪⎭,(2) and is norm bounded by one. Scalings can be included in 𝐶_𝑞 and 𝐵_𝑝, thus generalizing the bound. Δ_𝑘 can represent either a memoryless time-varying matrix with 𝜎(Δ_𝑖𝑘)≤1, for 𝑖=1,…,𝑡, 𝑘≥0, or the constraints:𝑝^𝑇_𝑖𝑘𝑝_𝑖𝑘≤𝑞^𝑇_𝑖𝑘𝑞_𝑖𝑘,𝑖=1,…,𝑡,(3) where 𝑝_𝑘=[𝑝_1𝑘,…,𝑝_𝑡𝑘]^𝑇, 𝑞_𝑘=[𝑞_1𝑘,…,𝑞_𝑡𝑘]^𝑇, and the partitioning is induced by Δ_𝑘. Each Δ_𝑘 is assumed to be either a full block or a repeated scalar block, and models a number of factors, such as dynamics or parameters, nonlinearities, that are unknown, unmodeled or neglected. In this work, we only consider full blocks for simplicity.

In terms of the state space matrices, this formulation can be viewed as replacing a fixed (𝐴,𝐵_𝑢,𝐵_𝑤) by (𝐴,𝐵_𝑢,𝐵_𝑤)∈(𝒜,ℬ_𝑢,ℬ_𝑤), where𝒜,ℬ_𝑢,ℬ_𝑤=𝐴+𝐵_𝑝Δ_𝑘𝐶_𝑞,𝐵_𝑢+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑢,𝐵_𝑤+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑤∣Δ_𝑘∈𝚫_𝑘.(4)

In robust model predictive control, we consider norm-bounded uncertainty and define stability in terms of quadratic stability [11] which requires the existence of a fixed quadratic Lyapunov function (𝑉(𝜁)=𝜁^𝑇𝑃𝜁, 𝑃>0) for all possible choices of the uncertainty parameters.

In the case of norm-bounded uncertainties:𝐴𝐵_𝑤𝐵_𝑢∈𝐴^𝑜𝐵^𝑜_𝑤𝐵^𝑜_𝑢+𝐹_𝐴Δ_𝐻𝐸_𝐴𝐸_𝑤𝐸_𝑢∶Δ∈𝚫,(5) where [𝐴^𝑜𝐵^𝑜_𝑤𝐵^𝑜_𝑢] represents the nominal model, Δ_𝐻=Δ(𝐼−𝐻Δ)⁻¹, withΔ∈𝚫∶=Δ=diag𝛿₁𝐼_𝑞1,…,𝛿_𝑙𝐼_𝑞𝑙,Δ_𝑙+1,…,Δ_𝑙+𝑓∶‖Δ‖≤1,𝛿_𝑖∈ℛ,Δ_𝑖∈ℛ^{𝑞𝑖×𝑞𝑖}(6) and where 𝐹_𝐴, 𝐸_𝐴, 𝐸_𝑤, 𝐸_𝑢, and 𝐻 are known and constant matrices with appropriate dimensions. This linear fractional representation of uncertainty, which is assumed to be well posed over Δ (i.e., det(𝐼−𝐻Δ)≠0 for all Δ∈Δ), has great generality and is used widely in robust control theory [12].

We use the following lemma, which is a slight modification of a result in [13] and which uses the fact that Δ∈Δ to remove explicit dependence on Δ for the solution with norm bounded uncertainties.

Lemma 1. Let Δ be as described in (6) and define the subspaces 𝚺=diag𝑆₁,…,𝑆_𝑙,𝜆₁𝐼_𝑞𝑙+1,…,𝜆_𝑠𝐼_{𝑞𝑙+𝑓}∶𝑆_𝑖=𝑆^𝑇_𝑖∈ℛ^{𝑞𝑖×𝑞𝑖},𝜆_𝑗∈ℛ,𝚪=diag𝐺₁,…,𝐺_𝑙,0_𝑞𝑙+1,…,0_{𝑞𝑙+𝑓}∶𝐺_𝑖=−𝐺^𝑇_𝑖∈ℛ^{𝑞𝑖×𝑞𝑖}.(7) Let 𝑇₁=𝑇^𝑇₁, 𝑇₂, 𝑇₃, 𝑇₄ be matrices with appropriate dimensions. We have det(𝐼−𝑇₄Δ)≠0 and 𝑇₁+𝑇₂Δ(𝐼−𝑇₄Δ)⁻¹𝑇₃+𝑇^𝑇₃(𝐼−Δ^𝑇𝑇^𝑇₄)⁻¹Δ^𝑇𝑇^𝑇₂<0 for every Δ∈Δ if there exist 𝑆∈𝚺 and 𝐺∈Γ such that 𝑆>0 and ⎡⎢⎢⎣𝑇₁+𝑇₂𝑆𝑇^𝑇₂𝑇^𝑇₃+𝑇₂𝑆𝑇^𝑇₄+𝑇₂𝐺𝑇₃+𝑇₄𝑆𝑇^𝑇₂+𝐺^𝑇𝑇^𝑇₂𝑇₄𝑆𝑇^𝑇₄+𝑇₄𝐺+𝐺^𝑇𝑇^𝑇₄−𝑆⎤⎥⎥⎦<0.(8) If Δ is unstructured, then (8) becomes ⎡⎢⎢⎣𝑇₁+𝜆𝑇₂𝑇^𝑇₂𝑇^𝑇₃+𝜆𝑇₂𝑇^𝑇₄𝑇₃+𝜆𝑇₄𝑇^𝑇₂𝜆𝑇₄𝑇^𝑇₄−𝐼⎤⎥⎥⎦<0,(9) for some scalar 𝜆>0. In this case, condition (9) is both necessary and sufficient.

We also use the following Schur complement result [14].

Lemma 2. Let 𝑋₁₁=𝑋^𝑇₁₁ and 𝑋₂₂=𝑋^𝑇₂₂. Then ⎡⎢⎢⎣𝑋₁₁𝑋₁₂𝑋^𝑇₁₂𝑋₂₂⎤⎥⎥⎦≥0⟺𝑋₂₂≥0,𝑋₂₂−𝑋^𝑇₁₂𝑋⁺₁₁𝑋₁₂≥0,𝑋₁₂𝐼−𝑋₁₁𝑋⁺₁₁=0,(10) where 𝑋⁺₁₁ denotes the Moore-Penrose pseudo-inverse of 𝑋₁₁.

We assume that the pair (𝐴,𝐵_𝑢) is stabilizable and that the disturbance is bounded as‖𝑤‖₂∶=⎷^∞_𝑘=0𝑤^𝑇_𝑘𝑤_𝑘≤𝑤,(11) where 𝑤≥0is know.

The aim is to find a state feedback control law {𝑢_𝑘=𝐹𝑥_𝑘} in ℒ₂, where 𝐹∈ℛ^{𝑛𝑢×𝑛}, such that the following constraints are satisfied for all Δ_𝑘∈Δ_𝑘.

(1) Closed-loop stability: the matrix 𝐴+𝐵_𝑢𝐹 is stable.

(2) Disturbance rejection: for given 𝛾>0, the transfer matrix from 𝑤 to 𝑧, denoted as 𝑇_𝑧𝑤, is quadratically stable and satisfies the ℋ_∞ constraint‖𝑧‖₂<𝛾‖𝑤‖₂,(12) for 𝑥₀=0.

(3) Regulation: for given 𝛼>0, the controlled output satisfies the ℋ₂ constraint:‖𝑧‖₂∶=⎷^∞_𝑘=0𝑧^𝑇_𝑘𝑧_𝑘<𝛼.(13)

(4) Input constraints: for given 𝐻₁,…,𝐻_𝑚𝑢∈ℛ^{𝑛𝑢×𝑛𝑢}, 𝐻_𝑗=𝐻^𝑇_𝑗≥0, ℎ₁,…,ℎ_𝑚𝑢∈ℛ^𝑛𝑢×1, and 𝑢₁,…,𝑢_𝑚𝑢∈ℛ, the inputs satisfy the quadratic constraints:𝑢^𝑇_𝑘𝐻_𝑗𝑢_𝑘+2ℎ^𝑇_𝑗𝑢_𝑘≤𝑢_𝑗,∀𝑘;for𝑗=1,…,𝑚_𝑢.(14)

(5) State/output constraints: for given 𝐺₁,…,𝐺_𝑚𝑥∈ℛ^𝑛×𝑛, 𝐺_𝑗=𝐺^𝑇_𝑗≥0, 𝑔₁,…,𝑔_𝑚𝑥∈ℛ^𝑛×1, and 𝑥₁,…,𝑥_𝑚𝑥∈ℛ the states/outputs satisfy the quadratic constraints:𝑥^𝑇_𝑘+1𝐺_𝑗𝑥_𝑘+1+2𝑔^𝑇_𝑗𝑥_𝑘+1≤𝑥_𝑗,∀𝑘;for𝑗=1,…,𝑚_𝑥.(15) An 𝐹∈ℛ^{𝑛𝑢×𝑛} satisfying these requirements will be called an admissible state feedback gain.

3. LMI Formulation of Sufficiency Conditions

The next theorem, which is the main result of this paper, derives sufficient conditions, in the form of LMIs, for the existence of an admissible 𝐹.

Theorem 3. Let all variables, definitions, and assumptions be as above. Then there exists an admissible state feedback gain matrix 𝐹 if there exists solutions 𝑄=𝑄^𝑇∈ℛ^𝑛×𝑛, 𝑌∈ℛ^{𝑛𝑢×𝑛}, 𝛿_𝑗≥0, 𝜇_𝑗≥0, 𝜈_𝑗≥0, Λ=diag(𝜆₁𝐼,…,𝜆_𝑡𝐼)>0, and Ψ_𝑗=diag(𝜓₁𝐼,…,𝜓_𝑡𝐼)>0 to the LMIs shown in (16)–(19).

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑄⋆⋆⋆⋆⋆⋆0−𝛼²𝛾²𝐼⋆⋆⋆⋆⋆00−Λ⋆⋆⋆⋆𝐴𝑄+𝐵_𝑢𝑌𝛼²𝐵_𝑤𝐵_𝑝Λ−𝑄⋆⋆⋆𝐶_𝑞𝑄+𝐷_𝑞𝑢𝑌𝛼²𝐷_𝑞𝑤00−Λ⋆⋆𝐶_𝑧𝑄0000−𝛼²𝐼⋆𝐷_𝑧𝑢𝑌00000−𝛼²𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(16)⎡⎢⎢⎢⎢⎣1⋆⋆𝛾²𝑤²𝛼²𝛾²𝑤²⋆𝑥₀0𝑄⎤⎥⎥⎥⎥⎦≥0,(17)⎡⎢⎢⎢⎢⎢⎢⎢⎣Q⋆⋆⋆𝐻^1/2_𝑗𝑌𝜇_𝑗𝐼⋆⋆−ℎ^𝑇_𝑗𝑌0𝜇_𝑗𝑢_𝑗⋆00𝜇_𝑗1⎤⎥⎥⎥⎥⎥⎥⎥⎦≥0,𝑗=1,…,𝑚_𝑢,(18)   ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑄⋆⋆⋆⋆⋆⋆0𝛿_𝑗𝐼⋆⋆⋆⋆⋆00Ψ_𝑗⋆⋆⋆⋆𝐺^1/2_𝑗𝐴𝑄+𝐵_𝑢𝑌𝜈_𝑗G^1/2_𝑗𝐵_𝑤𝐺^1/2_𝑗𝐵_𝑝Ψ_𝑗𝜈_𝑗𝐼⋆⋆⋆𝐶_𝑞𝑄+𝐷_𝑞𝑢𝑌𝜈_𝑗𝐷_𝑞𝑤00Ψ_𝑗⋆⋆−𝑔^𝑇_𝑗𝐴𝑄+𝐵_𝑢𝑌−𝜈_𝑗𝑔^𝑇_𝑗𝐵_𝑤−𝑔^𝑇_𝑗𝐵_𝑝Ψ_𝑗00𝜈_𝑗𝑥_𝑗−𝛿_𝑗𝑤²⋆00000𝜈_𝑗1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦≥0,𝑗=1,…,𝑚_𝑥.(19)

Here, ⋆ represents terms readily inferred from symmetry and the partitioning of Λ and Ψ_𝑗 is induced by the partitioning of Δ_𝑘. If such solutions exist, then 𝐹=𝑌𝑄⁻¹.

Remark 4. The variables in the LMI minimization of Theorem 3 are computed online at time 𝑘, the subscript 𝑘 is omitted for convenience.

Proof. Using 𝑢_𝑘=𝐹𝑥_𝑘, the dynamics in (1) become 𝑥_𝑘+1=(𝐴+𝐵_𝑢𝐹^{𝐴𝑐𝑙})𝑥_𝑘+𝐵_𝑤𝑤_𝑘+𝐵_𝑝𝑝_𝑘,𝑧_𝑘=^{𝐶𝑐𝑙}⎡⎢⎢⎣𝐶_𝑧𝐷_𝑧𝑢𝐹⎤⎥⎥⎦𝑥_𝑘.(20)
Consider a quadratic function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥, 𝑃>0 of the state 𝑥_𝑘. It follows from (20) that 𝑉𝑥_𝑘+1−𝑉𝑥_𝑘=𝑥^𝑇_𝑘𝐴^𝑇_𝑐𝑙𝑃𝐴_𝑐𝑙−𝑃𝑥_𝑘+𝑥^𝑇_𝑘𝐴^𝑇_𝑐𝑙𝑃𝐵_𝑤𝑤_𝑘+𝑥^𝑇_𝑘𝐴^𝑇_𝑐𝑙𝑃𝐵_𝑝𝑝_𝑘+𝑤^𝑇_𝑘𝐵^𝑇_𝑤𝑃𝐴_𝑐𝑙𝑥_𝑘+𝑤^𝑇_𝑘𝐵^𝑇_𝑤𝑃𝐵_𝑤𝑤_𝑘+𝑤^𝑇_𝑘𝐵^𝑇_𝑤𝑃𝐵_𝑝𝑝_𝑘+𝑝^𝑇_𝑘𝐵^𝑇_𝑝𝑃𝐴_𝑐𝑙𝑥_𝑘+𝑝^𝑇_𝑘𝐵^𝑇_𝑝𝑃𝐵_𝑤𝑤_𝑘+𝑝^𝑇_𝑘𝐵^𝑇_𝑝𝑃𝐵_𝑝𝑝_𝑘=𝑥^𝑇_𝑘𝑤^𝑇_𝑘𝑝^𝑇_𝑘𝐾⎡⎢⎢⎢⎢⎣𝑥_𝑘𝑤_𝑘𝑝_𝑘⎤⎥⎥⎥⎥⎦−𝑥^𝑇_𝑘𝐶^𝑇_𝑐𝑙𝐶_𝑐𝑙𝑥_𝑘+𝛾²𝑤^𝑇_𝑘𝑤_𝑘,(21) where 𝐾=⎡⎢⎢⎢⎢⎢⎣𝐴^𝑇_𝑐𝑙𝑃𝐴_𝑐𝑙−𝑃+𝐶^𝑇_𝑐𝑙𝐶_𝑐𝑙𝐴^𝑇_𝑐𝑙𝑃𝐵_𝑤𝐴^𝑇_𝑐𝑙𝑃𝐵_𝑝𝐵^𝑇_𝑤𝑃𝐴_𝑐𝑙𝐵^𝑇_𝑤𝑃𝐵_𝑤−𝛾²𝐼𝐵^𝑇_𝑤𝑃𝐵_𝑝𝐵^𝑇_𝑝𝑃𝐴_𝑐𝑙𝐵^𝑇_𝑝𝑃𝐵_𝑤𝐵^𝑇_𝑝𝑃𝐵_𝑝⎤⎥⎥⎥⎥⎥⎦.(22) Using 𝑞_𝑘=(𝐶_𝑞+𝐷_𝑞𝑢𝐹)𝑥_𝑘+𝐷_𝑞𝑤𝑤_𝑘, 𝑞^𝑇_𝑘Λ𝑞_𝑘=𝑥^𝑇_𝑘𝐶_𝑞+𝐷_𝑞𝑢𝐹^𝑇Λ𝐶_𝑞+𝐷_𝑞𝑢𝐹𝑥_𝑘+𝑥^𝑇_𝑘𝐶_𝑞+𝐷_𝑞𝑢𝐹^𝑇Λ𝐷_𝑞𝑤𝑤_𝑘+𝑤^𝑇_𝑘𝐷^𝑇_𝑞𝑤Λ𝐶_𝑞+𝐷_𝑞𝑢𝐹𝑥_𝑘+𝑤^𝑇_𝑘𝐷^𝑇_𝑞𝑤Λ𝐷_𝑞𝑤𝑤_𝑘,(23) where Λ=diag(𝜆₁𝐼,…𝜆_𝑡𝐼).
Substituting (23) into (21), it can be verified that we can write𝑉𝑥_𝑘+1−𝑉𝑥_𝑘=𝑥^𝑇_𝑘𝑤^𝑇_𝑘𝑝^𝑇_𝑘𝐾⎡⎢⎢⎢⎢⎣𝑥_𝑘𝑤_𝑘𝑝_𝑘⎤⎥⎥⎥⎥⎦+𝑝^𝑇_𝑘Λ𝑝_𝑘−𝑞^𝑇_𝑘Λ𝑞_𝑘−𝑥^𝑇_𝑘𝐶^𝑇_𝑐𝑙𝐶_𝑐𝑙𝑥_𝑘+𝛾²𝑤^𝑇_𝑘𝑤_𝑘,(24) where 𝐾 is defined in (25) and 𝐶_𝑝𝑤∶=𝐶_𝑞+𝐷_𝑞𝑢𝐹.𝐾=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴^𝑇_𝑐𝑙𝑃𝐴_𝑐𝑙−𝑃+𝐶^𝑇_𝑐𝑙𝐶_𝑐𝑙+𝐶^𝑇_𝑝𝑤Λ𝐶_𝑝𝑤𝐴^𝑇_𝑐𝑙𝑃𝐵_𝑤+𝐶^𝑇_𝑝𝑤Λ𝐷_𝑞𝑤𝐴^𝑇_𝑐𝑙𝑃𝐵_𝑝𝐵^𝑇_𝑤𝑃𝐴_𝑐𝑙+𝐷^𝑇_𝑞𝑤Λ𝐶_𝑝𝑤𝐵^𝑇_𝑤𝑃𝐵_𝑤−𝛾²𝐼+𝐷^𝑇_𝑞𝑤Λ𝐷_𝑞𝑤𝐵^𝑇_𝑤𝑃𝐵_𝑝𝐵^𝑇_𝑝𝑃𝐴_𝑐𝑙𝐵^𝑇_𝑝𝑃𝐵_𝑤𝐵^𝑇_𝑝𝑃𝐵_𝑝−Λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(25)
Assuming that lim_𝑘→∞𝑥_𝑘=0 we have^∞_𝑘=0𝑥^𝑇_𝑘+1𝑃𝑥_𝑘+1−𝑥^𝑇_𝑘𝑃𝑥_𝑘=−𝑥^𝑇₀𝑃𝑥₀.(26)
We write the ℋ₂ cost function as‖𝑧‖²₂=^∞_𝑘=0𝑥^𝑇_𝑘𝐶^𝑇_𝑐𝑙𝐶_𝑐𝑙𝑥_𝑘−𝛾²𝑤^𝑇_𝑘𝑤_𝑘+𝛾^2∞_𝑘=0𝑤^𝑇_𝑘𝑤_𝑘.(27)
Adding (26) and (27) and carrying out a simple manipulation gives‖𝑧‖²₂=𝑥^𝑇₀𝑃𝑥₀+𝛾²‖𝑤‖²₂+^∞_𝑘=0𝑥^𝑇_𝑘𝑤^𝑇_𝑘𝑝^𝑇_𝑘𝐾⎡⎢⎢⎢⎢⎣𝑥_𝑘𝑤_𝑘𝑝_𝑘⎤⎥⎥⎥⎥⎦+^∞_𝑘=0𝑝^𝑇_𝑘Λ𝑝_𝑘−𝑞^𝑇_𝑘Λ𝑞_𝑘,(28) where 𝐾 is defined in (25).
Setting 𝑥₀=0, it follows from (3), (12), and (28) that ‖𝑧‖₂<𝛾‖𝑤‖₂ if 𝐾<0 and Λ≥0. In this work, we will take Λ>0 to simplify our solution [8]. Using (2) and Lemma 1 it can be shown that 𝐾<0,(29) is also sufficient for quadratic stability of 𝑇_𝑧𝑤.
Next, we linearize the matrix inequality (29) by applying a Schur complement, to give ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃⋆⋆⋆⋆⋆⋆0−𝛾²𝐼⋆⋆⋆⋆⋆00−Λ⋆⋆⋆⋆𝐴_𝑐𝑙𝐵_𝑤𝐵_𝑝−𝑃⁻¹⋆⋆⋆𝐶_𝑝𝑤𝐷_𝑞𝑤00−Λ⁻¹⋆⋆𝐶_𝑧0000−𝐼⋆𝐷_𝑧𝑢𝐹00000−𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(30) Pre- and post-multiplying the equation above by diag(𝑃⁻¹,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼) gives ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑃⁻¹⋆⋆⋆⋆⋆⋆0−𝛾²𝐼⋆⋆⋆⋆⋆00−Λ⋆⋆⋆⋆𝐴_𝑐𝑙𝑃⁻¹𝐵_𝑤𝐵_𝑝−𝑃⁻¹⋆⋆⋆𝐶_𝑝𝑤𝑃⁻¹𝐷_𝑞𝑤00−Λ⁻¹⋆⋆𝐶_𝑧𝑃⁻¹0000−𝐼⋆𝐷_𝑧𝑢𝐹𝑃⁻¹00000−𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(31) setting 𝑄=𝛼²𝑃⁻¹, 𝐹=𝑌𝑃𝛼⁻²=𝑌𝑄⁻¹, 𝐶_𝑝𝑤=𝐶_𝑞+𝐷_𝑞𝑢𝐹 and multiplying through by 𝛼², the equation above becomes ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑄⋆⋆⋆⋆⋆⋆0−𝛼²𝛾²𝐼⋆⋆⋆⋆⋆00−𝛼²Λ⋆⋆⋆⋆𝐴𝑄+𝐵_𝑢𝑌𝛼²𝐵_𝑤𝛼²𝐵_𝑝−𝑄⋆⋆⋆𝐶_𝑞𝑄+𝐷_𝑞𝑢𝑌𝛼²𝐷_𝑞𝑤00−𝛼²Λ⁻¹⋆⋆𝐶_𝑧𝑄0000−𝛼²𝐼⋆𝐷_𝑧𝑢𝑌00000−𝛼²𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(32) Pre- and post-multiplying the equation above by diag(𝐼,𝐼,Λ⁻¹,𝐼,𝐼,𝐼,𝐼) gives ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝑄⋆⋆⋆⋆⋆⋆0−𝛼²𝛾²𝐼⋆⋆⋆⋆⋆00−𝛼²Λ⁻¹⋆⋆⋆⋆𝐴𝑄+𝐵_𝑢𝑌𝛼²𝐵_𝑤𝛼²𝐵_𝑝Λ⁻¹−𝑄⋆⋆⋆𝐶_𝑞𝑄+𝐷_𝑞𝑢𝑌𝛼²𝐷_𝑞𝑤00−𝛼²Λ⁻¹⋆⋆𝐶_𝑧𝑄0000−𝛼²𝐼⋆𝐷_𝑧𝑢𝑌00000−𝛼²𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(33) The equation above is a bilinear matrix inequality, thus by defining 𝛼²Λ⁻¹ as a variable Λ, we get the LMI in (16).
Now, it follows from (3), (11), (28), and (29) that, ‖𝑧‖²₂≤𝑥^𝑇₀𝑃𝑥₀+𝛾²‖𝑤‖²₂≤𝑥^𝑇₀𝑃𝑥₀+𝛾²𝑤².(34) Thus the ℋ₂ constraint in (13) is satisfied if 𝑥^𝑇₀𝑃𝑥₀+𝛾²𝑤²<𝛼².(35) Dividing by 𝛼², rearranging and using a Schur complement give (17) as an LMI sufficient condition for (13).
To turn (14) and (15) into LMIs, we first show that 𝑥^𝑇_𝑘𝑃𝑥_𝑘≤𝛼²∀𝑘>0. Since 𝐾<0, it follows from (3) and (24) that 𝑥^𝑇_𝑘+1𝑃𝑥_𝑘+1−𝑥^𝑇_𝑘𝑃𝑥_𝑘≤𝛾²𝑤^𝑇_𝑘𝑤_𝑘.(36) Applying this inequality recursively, we get 𝑥^𝑇_𝑘𝑃𝑥_𝑘≤𝑥^𝑇₀𝑃𝑥₀+𝛾^2𝑘−1_𝑗=0𝑤^𝑇_𝑗𝑤_𝑗≤𝑥^𝑇₀𝑃𝑥₀+𝛾²𝑤²<𝛼².(37) It follows that ‖‖𝑃^1/2𝑥_𝑘‖‖²<𝛼²,(38) or equivalently, 𝑥^𝑇_𝑘𝑄⁻¹𝑥_𝑘<1,∀𝑘>0.(39)
Next, we transform the constraints in (14) to a set of LMIs. Setting 𝐹=𝑌𝑄⁻¹=𝑌𝑃𝛼⁻² and 𝑢_𝑘=𝐹𝑥_𝑘, 𝑒_𝑗𝑢_𝑘∶=𝑢^𝑇_𝑘𝐻_𝑗𝑢_𝑘+2ℎ^𝑇_𝑗𝑢_𝑘−𝑢_𝑗=𝑥^𝑇_𝑘𝑄⁻¹𝑌^𝑇𝐻_𝑗𝑌𝑄⁻¹𝑥_𝑘+2ℎ^𝑇_𝑗𝑌𝑄⁻¹𝑥_𝑘−𝑢_𝑗.(40)
Now for any 𝜇_𝑗∈ℛ, we can write 𝑒_𝑗𝑢_𝑘=−𝜇_𝑗1−𝑥^𝑇_𝑘𝑄⁻¹𝑥_𝑘−⎡⎢⎢⎣𝑥_𝑘1⎤⎥⎥⎦^𝑇×⎛⎜⎜⎝⎡⎢⎢⎣𝜇_𝑗𝑄⁻¹−𝑄⁻¹𝑌^𝑇𝐻_𝑗𝑌𝑄⁻¹−𝑄⁻¹𝑌^𝑇ℎ_𝑗−ℎ^𝑇_𝑗𝑌𝑄⁻¹−𝜇_𝑗+𝑢_𝑗⎤⎥⎥⎦⎞⎟⎟⎠×⎡⎢⎢⎣𝑥_𝑘1⎤⎥⎥⎦.(41)
Therefore a sufficient condition for e_𝑗(𝑢_𝑘)≤0 is 𝜇_𝑗≥0 and ⎡⎢⎢⎣𝜇_𝑗𝑄⁻¹−𝑄⁻¹𝑌^𝑇𝐻_𝑗𝑌𝑄⁻¹−𝑄⁻¹𝑌^𝑇ℎ_𝑗−ℎ^𝑇_𝑗𝑌𝑄⁻¹𝑢_𝑗−𝜇_𝑗⎤⎥⎥⎦≥0.(42)
Pre- and post-multiplying by diag(𝑄,𝐼) gives a bilinear matrix inequality and applying a Schur complement, we get ⎡⎢⎢⎢⎢⎢⎣𝜇_𝑗𝑄𝑌^𝑇𝐻^1/2_𝑗−𝑌^𝑇ℎ_𝑗𝐻^1/2_𝑗𝑌𝐼0−ℎ^𝑇_𝑗𝑌0𝑢_𝑗−𝜇_𝑗⎤⎥⎥⎥⎥⎥⎦≥0.(43) Pre- and post-multiplying the above bilinear matrix inequality by diag(𝜇^−1/2_𝑗,𝜇^1/2_𝑗,𝜇^1/2_𝑗) and applying a Schur complement, this is equivalent to the LMI in (18).
Finally, to obtain an LMI formulation of the state constraints (15), the following analogous steps are carried out: 𝑓_𝑗𝑥_𝑘+1∶=⎡⎢⎢⎢⎢⎣𝑥_𝑘𝑤_𝑘𝑝_𝑘⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎢⎣𝐴^𝑇_𝑐𝑙𝐵^𝑇_𝑤𝐵^𝑇_𝑝⎤⎥⎥⎥⎥⎦𝐺_𝑗𝐴_𝑐𝑙𝐵_𝑤𝐵_𝑝⎡⎢⎢⎢⎢⎣𝑥_𝑘𝑤_𝑘𝑝_𝑘⎤⎥⎥⎥⎥⎦+2𝑔^𝑇_𝑗𝐴_𝑐𝑙𝐵_𝑤𝐵_𝑝⎡⎢⎢⎢⎢⎣𝑥_𝑘𝑤_𝑘𝑝_𝑘⎤⎥⎥⎥⎥⎦−𝑥_𝑗.(44) Now for any 𝜈_𝑗,𝜌_𝑗∈ℛ, we can write^{𝑓𝑗𝑥𝑘+1=−𝜈𝑗1−𝑥𝑇𝑘𝑄−1𝑥𝑘−𝜌𝑗𝑤2−𝑤𝑇𝑘𝑤𝑘}₋^⎡⎢_⎢⎢⎢⎣^𝑥𝑘_𝑤𝑘1^⎤⎥_⎥⎥⎥⎦^𝑇⎛_{⎜⎜⎜⎜⎝}^⎡⎢_⎢⎢⎢⎣^{𝜈𝑗𝑄−1−𝐴𝑐𝑙+𝐵𝑝Δ𝑘𝐶𝑝𝑤𝑇𝐺𝑗𝐴𝑐𝑙+𝐵𝑝Δ𝑘𝐶𝑝𝑤⋆⋆}_{−𝐵𝑤+𝐵𝑝Δ𝑘𝐷𝑞𝑤𝑇𝐺𝑗𝐴𝑐𝑙+𝐵𝑝Δ𝑘𝐶𝑝𝑤𝜌𝑗𝐼⋆−𝑔𝑇𝑗𝐴𝑐𝑙+𝐵𝑝Δ𝑘𝐶𝑝𝑤−𝑔𝑇𝑗𝐵𝑤+𝐵𝑝Δ𝑘𝐷𝑞𝑤−𝜈𝑗−𝜌𝑗𝑤2+𝑥𝑗}^⎤⎥_⎥⎥⎥⎦^⎞_{⎟⎟⎟⎟⎠}^⎡⎢_⎢⎢⎢⎣^𝑥𝑘_𝑤𝑘1^⎤⎥_{⎥⎥⎥⎦.}(45)Therefore a sufficient condition for 𝑓_𝑗(𝑥_𝑘+1)≥0 is 𝜈_𝑗≥0, 𝜌_𝑗≥0 and⎡⎢⎢⎢⎢⎣𝜈_𝑗𝑄⁻¹⋆⋆0𝜌_𝑗𝐼⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙+𝐵_𝑝Δ_𝑘𝐶_𝑝𝑤−𝑔^𝑇_𝑗𝐵_𝑤+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑤𝑥_𝑗−𝜈_𝑗−𝜌_𝑗𝑤²⎤⎥⎥⎥⎥⎦−⎡⎢⎢⎢⎢⎣𝐴_𝑐𝑙+𝐵_𝑝Δ_𝑘𝐶_𝑝𝑤^𝑇𝐵_𝑤+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑤^𝑇0⎤⎥⎥⎥⎥⎦𝐺_𝑗𝐴_𝑐𝑙+𝐵_𝑝Δ_𝑘𝐶_𝑝𝑤𝐵_𝑤+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑤0≥0.(46)Applying Schur complement to the above equation, we get⎡⎢⎢⎢⎢⎢⎢⎢⎣𝜈_𝑗𝑄⁻¹⋆⋆⋆0𝜌_𝑗𝐼⋆⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙+𝐵_𝑝Δ_𝑘𝐶_𝑝𝑤−𝑔^𝑇_𝑗𝐵_𝑤+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑤𝑥_𝑗−𝜈_𝑗−𝜌_𝑗𝑤²⋆𝐺^1/2_𝑗𝐴_𝑐𝑙+𝐵_𝑝Δ_𝑘𝐶_𝑝𝑤𝐺^1/2_𝑗𝐵_𝑤+𝐵_𝑝Δ_𝑘𝐷_𝑞𝑤0𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎦≥0.(47)
When Δ is structured we proceed as follows. For norm-bounded uncertainty, we first separate the terms involving modeling uncertainties from the other terms as ⎡⎢⎢⎢⎢⎢⎢⎢⎣𝜈_𝑗𝑄⁻¹⋆⋆⋆0𝜌_𝑗𝐼⋆⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙−𝑔^𝑇_𝑗𝐵_𝑤𝑥_𝑗−𝜈_𝑗−𝜌_𝑗𝑤²⋆−𝐺^1/2_𝑗𝐴_𝑐𝑙−𝐺^1/2_𝑗𝐵_𝑤0𝐼⎤⎥⎥⎥⎥⎥⎥⎥_{−𝑇1}⎦+⎡⎢⎢⎢⎢⎢⎢⎢⎣00−𝑔^𝑇_𝑗𝐵_𝑝𝐺^1/2_𝑗𝐵_𝑝⎤⎥⎥⎥⎥⎥⎥⎥_{−𝑇2}⎦Δ_𝑘𝐶_𝑝𝑤𝐷_𝑞𝑤00_{𝑇3}+⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐶^𝑇_𝑝𝑤𝐷^𝑇_𝑞𝑤00⎤⎥⎥⎥⎥⎥⎥⎥_{𝑇𝑇3}⎦Δ^𝑇_𝑘00−𝐵^𝑇_𝑝𝑔_𝑗𝐵^𝑇_𝑝𝐺^1/2_{−𝑇𝑇2}≥0.(48) Equation (48) is equivalent to −𝑇₁−𝑇₂Δ𝑇₃−𝑇^𝑇₃Δ^𝑇𝑇^𝑇₂>0, where 𝑇₄=0. By using (8) from Lemma 1, we have ⎡⎢⎢⎣−𝑇₁−𝑇^𝑇₃𝑆𝑇₃−𝑇₂−𝑇^𝑇₂𝑆⎤⎥⎥⎦>0.(49) Applying Schur complement to (49), we get ⎡⎢⎢⎢⎢⎣−𝑇₁𝑇^𝑇₃−𝑇₂𝑇₃𝑆⁻¹0−𝑇^𝑇₂0𝑆⎤⎥⎥⎥⎥⎦>0.(50) Substituting the variables from (48) into (50) and swapping the third and sixth diagonal elements, we get ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜈_𝑗𝑄⁻¹⋆⋆⋆⋆⋆0𝜌_𝑗𝐼⋆⋆⋆⋆00𝑆⋆⋆⋆𝐺^1/2_𝑗𝐴_𝑐𝑙𝐺^1/2_𝑗𝐵_𝑤𝐺^1/2_𝑗𝐵_𝑝𝐼⋆⋆𝐶_𝑝𝑤𝐷_𝑞𝑤00𝑆⁻¹⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙−𝑔^𝑇_𝑗𝐵_𝑤−𝑔^𝑇_𝑗𝐵_𝑝00𝑥_𝑗−𝜈_𝑗−𝜌_𝑗𝑤²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦≥0.(51) Pre- and post-multiplying by diag(𝑄,𝐼,𝐼,𝐼,𝐼,𝐼) gives ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜈_𝑗𝑄⋆⋆⋆⋆⋆0𝜌_𝑗𝐼⋆⋆⋆⋆00𝑆⋆⋆⋆𝐺^1/2_𝑗𝐴_𝑐𝑙𝐺^1/2_𝑗𝐵_𝑤𝐺^1/2_𝑗𝐵_𝑝𝐼⋆⋆𝐶_𝑝𝑤𝐷_𝑞𝑤00𝑆⁻¹⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙−𝑔^𝑇_𝑗𝐵_𝑤−𝑔^𝑇_𝑗𝐵_𝑝00𝑥_𝑗−𝜈_𝑗−𝜌_𝑗𝑤²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦≥0.(52) The above equation is bilinear and thus we pre- and post-multiply it by diag(𝜈^−1/2_𝑗,𝜈^1/2_𝑗,𝜈^1/2_𝑗,𝜈^1/2_𝑗,𝜈^1/2_𝑗,𝜈^1/2_𝑗) to obtain⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑄⋆⋆⋆⋆⋆0𝜈_𝑗𝜌_𝑗𝐼⋆⋆⋆⋆00𝜈_𝑗𝑆⋆⋆⋆𝐺^1/2_𝑗𝐴_𝑐𝑙𝜈_𝑗𝐺^1/2_𝑗𝐵_𝑤𝜈_𝑗𝐺^1/2_𝑗𝐵_𝑝𝜈_𝑗𝐼⋆⋆𝐶_𝑝𝑤𝜈_𝑗𝐷_𝑞𝑤00𝜈_𝑗𝑆⁻¹⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙−𝜈_𝑗𝑔^𝑇_𝑗𝐵_𝑤−𝜈_𝑗𝑔^𝑇_𝑗𝐵_𝑝00𝜈_𝑗𝑥_𝑗−𝜈²_𝑗−𝜈_𝑗𝜌_𝑗𝑤²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦≥0.(53)From the above equation, we can see that the variable 𝑆 and its inverse appear in the matrix inequality, thus to make it uniform, we pre- and post-multiply by diag(𝐼,𝐼,𝑆⁻¹,𝐼,𝐼,𝐼) to get⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑄⋆⋆⋆⋆⋆0𝜈_𝑗𝜌_𝑗𝐼⋆⋆⋆⋆00𝜈_𝑗𝑆⁻¹⋆⋆⋆𝐺^1/2_𝑗𝐴_𝑐𝑙𝜈_𝑗𝐺^1/2_𝑗𝐵_𝑤𝜈_𝑗𝐺^1/2_𝑗𝐵_𝑝𝑆⁻¹𝜈_𝑗𝐼⋆⋆𝐶_𝑝𝑤𝜈_𝑗𝐷_𝑞𝑤00𝜈_𝑗𝑆⁻¹⋆−𝑔^𝑇_𝑗𝐴_𝑐𝑙−𝜈_𝑗𝑔^𝑇_𝑗𝐵_𝑤−𝜈_𝑗𝑔^𝑇_𝑗𝐵_𝑝𝑆⁻¹00𝜈_𝑗𝑥_𝑗−𝜈²_𝑗−𝜈_𝑗𝜌_𝑗𝑤²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦≥0.(54)Thus the above equation is a nonlinear matrix inequality in 𝜈²_𝑗 and bilinear in 𝜈_𝑗𝜌_𝑗 and 𝜈_𝑗𝑆⁻¹, hence we define new variables Ψ_𝑗=𝜈_𝑗𝑆⁻¹ and 𝛿_𝑗=𝜈_𝑗𝜌_𝑗 and finally applying a Schur complement, we obtain the LMI of (19).

Remark 5. The input and state constraints used in this paper are more general than those used in [9] in that we allow linear terms and so this makes it possible to include asymmetric or hyperplane constraints.

Remark 6. When there is no uncertainty, the problem reduces to disturbance rejection technique considered in [9].

Remark 7. When there is no disturbance, the results reduce to those of [7].

Remark 8. The method used in this paper guarantees recursive feasibility (see [15, Chapter  4]). Also see [16] for a different approach.

4. Numerical Examples

In this section, we present two examples that illustrate the implementation of the proposed scheme. In the first example we consider a solenoid system, and in the second example we consider the coupled spring-mass system. The solution to the linear objective minimization was computed using LMI Control Toolbox in the MATLAB^® environment and 𝛼² was set as a variable.

4.1. Example 1

We consider a modified version of the solenoid system adapted from [17]. The system (see Figure 1) consists of a central object wrapped with coil and is attached to a rigid surface via a spring and damper, which forms a passive vibration isolator. The solenoid is one of the common actuator components. The basic principle of operation involves a moving ferrous core (a piston) that moves inside a wire coil. Normally, the piston is held outside the core by a spring and damper. When a voltage is applied to the coil and current flows, the coil builds up a magnetic field that attracts the piston and pulls it into the center of the coil. The piston can be used to supply a linear force. Application of this includes pneumatic valves and car door openers.

Figure 1 

Solenoid system.

The system is modeled by ⎡⎢⎢⎣𝑥¹_𝑘+1𝑥²_𝑘+1⎤⎥⎥⎦=⎡⎢⎢⎣0.61480.0315−0.3155−0.0162⎤⎥⎥⎦⎡⎢⎢⎣𝑥¹_𝑘𝑥²_𝑘⎤⎥⎥⎦+⎡⎢⎢⎣0.03850.0315⎤⎥⎥⎦𝑢_𝑘+⎡⎢⎢⎣0.003850.00315⎤⎥⎥⎦𝑤_𝑘+⎡⎢⎢⎣010⎤⎥⎥⎦𝑝_𝑘,𝑞_𝑘=𝐶_𝑞𝑥_𝑘+𝐷_𝑞𝑢𝑢_𝑘+𝐷_𝑞𝑤𝑤_𝑘,𝑝_𝑘=Δ𝑞_𝑘,𝑧_𝑘=⎡⎢⎢⎣𝐶_𝑧𝑥_𝑘𝐷_𝑧𝑢𝑢_𝑘⎤⎥⎥⎦,(55) where 𝐶_𝑞=10,𝐷_𝑞𝑢=1,𝐷_𝑞𝑤=0,(56) where 𝑥¹ and 𝑥² are the position and the velocity of the plate. The cost function is specified using 𝐶_𝑧=diag(1,1) and 𝐷_𝑧𝑢=10. The magnetic force 𝑢 is the control variable, and 𝑤 is the external disturbance to the system, which is bounded in the range [−1,1]. The initial state is given as 𝑥₀=[10]^𝑇.

We choose 𝛾²=0.01 and 𝛾²=1. Figures 2 and 3 compare the closed-loop response for the high and low disturbance rejection levels, respectively, for randomly generated Δ’s. The optimization is feasible, the response is stable, and the performance is good. A control constraint of |𝑢_𝑘|≤0.5 is imposed, which is satisfied. The computation time for 100 samples was about 10 s, making 0.1 s per sample.

(a)

(b)

Figure 2 

Closed-loop response of the solenoid system with 𝛾2=0.01.

(a)

(b)

Figure 3 

Closed-loop response of the solenoid system with 𝛾2=1.

4.2. Example 2

We revisit a modified version of Example 2 reported in [7]. The system consists of a two-mass-spring model whose discrete-time equivalent is obtained using Euler first-order approximation with a sampling time of 0.1 s. The model in terms of disturbance and perturbation variables is𝑥_𝑘+1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣100.100100.1−0.1𝐾𝑚₁0.1𝐾𝑚₁100.1𝐾𝑚₂−0.1𝐾𝑚₂10⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑥_𝑘+⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣000.1𝑚₁0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑢_𝑘+𝐵_𝑤𝑤_𝑘+𝐵_𝑝𝑝_𝑘,𝑞_𝑘=𝐶_𝑞𝑥_𝑘+𝐷_𝑞𝑢𝑢_𝑘+𝐷_𝑞𝑤𝑤_𝑘,𝑝_𝑘=Δ𝑞_𝑘,𝑦_𝑘=0100𝑥_𝑘,(57) where 𝐵_𝑤=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣00.0100⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐵_𝑝=⎡⎢⎢⎢⎢⎢⎢⎣00−0.10.1⎤⎥⎥⎥⎥⎥⎥⎦,𝐶_𝑞=0.475−0.47500,𝐷_𝑞𝑤=0,𝐷_𝑞𝑢=0,(58) where 𝑥₁ and 𝑥₂ are the positions of body 1 and 2, and 𝑥₃ and 𝑥₄ are their velocities, respectively. 𝑚₁ and 𝑚₂ are the masses of the two bodies and 𝐾 is the spring constant. The initial state is given as 𝑥₀=[1100]^𝑇. The cost function is specified using 𝐶_𝑍=diag(1,1,1,1), and 𝐷_𝑧𝑢=1. We consider the system with 𝑚₁=𝑚₂=1 and 𝐾∈[0.5,10].

A persistent disturbance of the form 𝑤_𝑖=0.1 for all sample time was considered. Here we set 𝛾²=0.45 and 𝛾²=4. Figures 4 and 5 compare the closed-loop response for the high and low disturbance rejection levels, respectively, for randomly generated Δ’s. The value of 𝛾² for high disturbance rejection was the lowest value for which a feasible solution exists. An input constraint of |𝑢_𝑘|≤1 is imposed, which is satisfied. The computation time for 300 samples was about 47 s, making 0.16 s per sample.

(a)

(b)

Figure 4 

Closed-loop response of the coupled spring-mass system with 𝛾2=0.45.

(a)

(b)

Figure 5 

Closed-loop response of the coupled spring-mass system with 𝛾2=4.

4.3. Discussion

Note that the performance and response of the systems based on the high disturbance rejection level were better than those obtained using low disturbance rejection level, since the states and control are regulated to smaller steady state values. Constraints on the input were satisfied in both cases; however, the constraints were more conservative with respect to the control signal for the low disturbance rejection level. For example in the solenoid system, the control signal for high disturbance rejection level was 0.4826 and that for low disturbance rejection level was 0.4144. The issue of conservativeness in the mixed ℋ₂/ℋ_∞ setting has been considered in [18]. For the systems considered, the upper bound on the ℋ₂ cost function 𝛼² is depicted in Figures 6 and 7 for the high and low disturbance rejection levels. It can been seen that the performance coefficient obtained for the high and low disturbance rejection levels is small. However, the offset level on the low disturbance level is higher than that for the high disturbance level. This is due to the higher value of 𝛾² in 𝑥^𝑇₀𝑃𝑥₀+𝛾²𝑤². In Example 2, we have considered uncertainty in the model by using variable spring constant 𝐾.

(a) System with 𝛾2=0.01

(b) System with 𝛾2=1

Figure 6 

Upper bound on ℋ2 cost function for solenoid system.

(a) System with 𝛾2=0.45

(b) System with 𝛾2=4

Figure 7 

Upper bound on ℋ2 cost function for coupled spring-mass system.

5. Conclusion

In this paper, we proposed a robust model predictive control design technique using mixed ℋ₂/ℋ_∞ for time invariant discrete-time linear systems subject to constraints on the inputs and states. This method takes account of disturbances naturally by imposing the ℋ_∞-norm constraint in (14) and thus extends the work in [9] by the introduction of structured, norm-bounded uncertainty. The uncertain system was represented by LFTs. The development is based on full state feedback assumption and the on-line optimization involves the solution of an LMI-based linear objective minimization (convex optimization). Hence, the resulting state-feedback control law minimizes an upper bound on the robust objective function. The new approach reduces to that of [9] when there are no perturbations present in the system and to [7] when there are no disturbances. Thus, we have been able to show that it is possible to handle uncertainty and disturbance in the mixed ℋ₂/ℋ_∞ model predictive control design approach. The two examples illustrate the application of the proposed method.

Acknowledgments

This was parts supported by Commonwealth Scholarship Commission in the United Kingdom (CSCUK) under Grant NGCS-2004-258. With regard to this paper, particular thanks go to Dr. Imad M. Jaimoukha and Dr. Eric Kerrigan.