Measurement feedback controllers with constraints and their relation to the solution of Hamilton Jacobi inequalities

In this paper we propose a separation result for the stabilization of systems with control and measurement contraints, which clarifies the connection between the solution of a couple of Hamilton Jacobi inequalities (HJI) and the constraints imposed on the control and the estimation error fed back in the control loop. Once a solution of these HJI has been found a measurement feedback controller can be directly implemented. This controller has different features with respect to classical controllers: in classical control schemes an observer consists of a “copy” of the system plus a term proportional to the error between the actual measurement and the “estimated” measurement. Here, we allow a term which is a nonlinear function of this error. I. COMPLEX DYNAMICS AS INTERCONNECTION OF SIMPLER DYNAMICS Consider Σ : ẋ1 = x2 + wx1x3, ẋ2 = x3 + x1x 2 2 + w, ẋ3 = x4, μ = col(x1 + w, x2 + w, x3) with state x =col(x1, x2, x3), control input x4, measurements μ and exogenous disturbance w ∈ L∞[0,∞) with known ‖w‖∞ (L∞[0,∞) is the space of continuous and bounded functions over [0,∞) and ‖ · ‖∞ denotes the sup norm). We can consider Σ as the interconnection of three one– dimensional systems: Σ1 : ẋ1 = x2 +wx1x3, μ1 = x1 +w with state x1, control input x2 and measurement μ1; Σ2 : ẋ2 = x3 + x1x2 + w, μ2 = x2 + w with state x2, control input x3 and measurement μ2; Σ3 : ẋ3 = x4, μ3 = x3 with state x3, control input x4 and measurement μ3. Note that x2 (resp. x3) is a control input for Σ1 (resp. Σ2) but not for Σ: we say that x2 and x3 are virtual inputs of Σ. A virtual input cannot be used as control input for Σ; on the other hand, x4 is a control input both for Σ and Σ3; x3 and w are exogenous inputs for Σ1 while x1 and w are exogenous inputs for Σ2. An exogenous input of a system can be considered as an “exogenous disturbance” affecting the system itself and as such it cannot be manipulated to control the system. Thus, it is natural to distinguish the inputs of a system into control (or endogenous) inputs v and exogenous inputs χ. We denote the inputs altogether, endogenous and exogenous, by א; the exogenous input x3 of Σ1 is also a measurement of Σ3. An exogenous input of a system may be as well available as a feedback signal although not being a measurement of the system itself, since it is available as a measurement from other systems. Thus, it is natural to distinguish the measurements of a system into endogenous measurements μ (available from the system itself) and exogenous measurements ν (available from other systems). We denote the measurements altogether, endogenous and exogenous, by ς; the exogenous inputs χ2 =col(x1, w) of Σ2 are not exogenous measurements of Σ2. However, the exogenous inputs χ2 and the exogenous measurement μ1 of Σ2 satisfy the constraint V : μ1 = x1 + w. A system Σ with states x ∈ IR, inputs א ∈ IR and exogenous measurements ν ∈ IR satisfies a constraint V among x,א and ν or (x,א, ν) ∈ V if x,א, ν : V(x,א, ν) ≤ 0 for some continuous function V : IR × IR × IR → IR. This definition includes differential constraints as well and the case in which an exogenous input of a system is also an exogenous measurement of the system itself; the measurement equation μ1 = x1+w of Σ1 determines another type of constraint which has to satisfied by the state x1, the endogenous measurements μ1 and the exogenous input w of Σ1. A system Σ with states x ∈ IR, inputs א ∈ IR and measurements μ ∈ IR satisfies a constraint Y among x,א and μ or (x,א, μ) ∈ Y if x,א, μ : Y(x,א, μ) ≤ 0 for some continuous function Y : IR × IR × IR → IR. This definition includes trivially the case in which the state of a system is also an endogenous measurement of the system itself. We denote constraints V and Y altogether by M and we say (x,א, ς) ∈ M; the system Σ1 can be seen as a linear nominal system: ẋ1 = x2, μ1 = x1, with state uncertainty wx1x3 and measurement uncertainty w. Similar remarks can be repeated for Σ2 and Σ3. State and measurement uncertainty can be accomodated into a vector Ψ, which denotes the “uncertainty” of a system. The uncertainty can be seen as a continuous function Ψ : IR × IR × IR → IR, (x,א, t) → Ψ(x,א, t), such that Ψ(0, 0, t) = 0 for all t (only piecewise continuity w.r.t. t may be as well assumed). It is important for the controller design to have an upper bound of the incremental rates of the uncertainty with respect to the state and the inputs of the system. This allows to evaluate quantitatively the effect of the uncertainty Ψ(x,א, t) on the system through the state and the control input, on one hand, and the exogenous inputs, on the other. In the case of Σ1 with state x1, measurements ς1 (endogenous μ1 and exogenous ν1 = μ3), inputs א1 (control x2 and exogenous χ1 = col(x3, w)), uncertainty Ψ1(x1,א1) = col(wx1x3, w) and constraints M1 = (V1,Y1), with Y1 : μ1 = x1 + w and V1 : μ3 = x3, we have (‖Ψ1(x1,א1)‖−‖Ψ1|x1=0‖) ≤ 2x1‖w‖∞μ3(μ1 + ‖w‖∞) + w for all (x1,א1, ς1) ∈ M1. Note that the functions γ1 x1 (ς1) = 2‖w‖∞μ3(μ̃1 +‖w‖∞) and γ1 w = 1 give, respectively, an upper bound of the “incremental terms” (‖Ψ1(x1,א1)‖ − ‖Ψ1|x1=0‖)/x1 and (‖Ψ1|x1=0‖− ‖Ψ1|x1,w=0‖)/w under the constraints M1. Let IR≥ denote the set of nonnegative real numbers and IR the set of positive real numbers. We say that a system Σ with states x ∈ IR, inputs א ∈ IR and measurements ς ∈ IR, uncertainty Ψ(x,א, t) ∈ IR and constraints M has smooth incremental (from zero) rate γ z if there exist a nonempty subset {z} of {x,א} and a smooth function γ z : IR n × IR × IR → IR≥ such that (‖Ψ(x,א, t)‖−‖Ψ|z=0‖) ≤ γ z (x, χ, ς)‖z‖2 for all (x,א, ς) ∈ M and ∀t. Since f(s1, . . . , sq) = (f(s1, . . . , sq) − f |s1=0) + · · · + (f |s1,...,sq−2=0 − f |s1,...,sq−1=0) + f |s1,...,sq−1=0 for any function f : IR → IR such that f(0, . . . , 0) = 0, for a system Σ we can reasonably expect to have γ(ς)‖Ψ(x,א, t)‖ ≤ γ x (x, ς)‖x‖ + γ v (ς)‖v‖ + ∑ j∈I γ χj (x, χ, ς)χ 2 j , ∀t, (x,א, ς) ∈ M (1) where χj is the j–th element of χ, j ∈ I a given set of indeces, and γ x : IR n × IR → IR≥, γ v : IR → IR≥ and γ χj : IR n × IR × IR → IR≥ are smooth incremental rates (“rescaled” by the square of a smooth function γ : IR → IR). We say that a system Σ with states x ∈ IR, inputs א ∈ IR and measurements ς ∈ IR, uncertainty Ψ(x,א, t) ∈ IR and constraints M has smooth incremental rates γ x , γ γΨ v and γ γΨ χj , j ∈ I a given set of indeces, if (1) holds for some smooth functions γ x : IR × IR → IR≥, γ χj : IR × IR × IR → IR≥, j ∈ I, and γ v , γ : IR → IR. In this paper, using the general framework proposed above and further developing the preliminary results of [6], we propose a separation result into state feedback and observer design with control and state estimation error constraints. On this separation result it is possible to base a powerful control strategy which consists essentially of the following steps: split a n–dimensional system Σ into one–dimensional systems Σj , j = 1, . . . , n, each one with a state, a control input and a measurement; find a one– dimensional measurement feedback controller Cj for each one–dimensional system Σj according to the separation result; finally, take Cj , j = 1, . . . , n, as controller C for Σ (see [4] for details). II. A SEPARATION RESULT Consider a system Σ(x,א, ς,Ψ,M,U ,F) : ẋ = Ax + B2v + B1(ς)Ψ(x,א, t) μ = C2x + C1(ς)Ψ(x,א, t), x(t0) = x0 (2) with smooth B1(ς) and C1(ς) such that C 1 (ς)B1(ς) = 0 and R2(ς) := C1(ς) C 1 (ς) > 0 for all ς, states x ∈ IR, inputs א (control v ∈ IR and exogenous inputs χ ∈ IR), measurements ς (endogenous measurement μ ∈ IR and exogenous measurements ν ∈ IR), uncertainty Ψ, constraints M = (V,Y) and smooth incremental rates γ x , γ γΨ v and γ γΨ χj , j = 1, . . . , r (χj denotes as usual the j–th element of χ), a family U(t0) of continuous functions v : [t0, T ) → IR and a family F(t0) = N (t0) × X (t0) of continuous pairs (ν(t), χ(t)) with ν : [t0, T ) → IR and χ : [t0, T ) → IR, where T ∈ [0,∞]. We assume that Ψ(x,א, t) satisfies standard assumptions for the local existence and uniqueness of the trajectories of (2). Moreover, for any component νi(t) and χi(t) of ν(t) and, respectively, χ(t) wherever necessary from the context we assume that it is continuously differentiable over its domain of definition and satisfies a differentiable constraint of the form ν̇i(t) = Λi(x(t),א(t), t) and, respectively, χ̇i(t) = Λi(x(t),א(t), t), which as such will be included in the constraints V. By trajectory of (2) we mean the 4– tuple (x(t), v(t), ν(t), χ(t)), with (x(t),א(t), ς(t)) ∈ M(t) (i.e. satisfying the constraints M as functions of time), where v ∈ U(t0), (ν, χ) ∈ F(t0) and x(t) is the solution of (2) over its right maximal domain of definition. A starting point of this paper is to adopt for the controller design of (2) some type of certainty equivalence principle: first, find a state feedback controller by assuming full state information, i.e. μ = x, then design an observer to estimate on line the state x(t) and replace x(t) with the estimate σ(t) in the state feedback controller. Let IR 0 be the vector space IR s except the origin and denote √ vTNv by ‖v‖N for any vector v ∈ IR and symmetric positive semidefinite matrix N ∈ IRs×s. With full state information and according to a H∞ control strategy ([5]), the problem is to find smooth functions F : IRn×IRs+1 → IR and V : IRn×IRs → IR≥, (x, ν) → V (x, ν) := Vf (‖x‖Pf (ν)), proper and positive definite for each ν ∈ S ⊂ IR 0 with Pf (ν) symmetric and positive definite for each ν ∈ S, such that along the trajectories of (2) with v = F (x, ς) V̇f + γ x (x, ς)‖x‖ + γ v (ς)‖v‖