Both the McFarlane-Glover method (Section 4.2.2) and the mixed sensitivity method (Section 4.2.3) require to find controllers that minimize the norm of a weighted closed-loop interconnection. This section very briefly reviews some of the relevant synthesis machinery for obtaining such controllers. While Section 4.2 described how meaningful performance specifications can be formulated in terms of sensitivity functions, i. e., how a controller can bedesigned, the termsynthesisrefers to the technicality of obtaining a controller that achieves these performance specifications.
Throughout this section, the partitioned state space realization
G:
for the (open-loop)generalized plantis used. The input-output mapw→z describes the mixed sensitivity requirements for a closed-loop interconnection withu=K v. Thus,v de-notes the measured signals available to the controller (including references for feedforward) andudenotes available control inputs. Clearly, the pair (A, B2) must be stabilizable and (A, C2) must be detectable, for a stabilizing controller to exist.
A useful special structure can be achieved through loop-shifting and scalings under mild conditions [Safonov & Chiang 1989]. Specifically, it is possible to makeD22 = 0, D21 = [0 I] and D12 = [0 I]T. Further, denote C1T = [CT11C12T ], B1 = [B11B12], and D11 = [D11•1D11•2] =D111•
D112•
=D1111 D1112
D1121 D1122
such that the partitioning is compatible.
The state space realization (4.18) then takes the form
G: andu→z2. The implied rank conditions have clear interpretations in terms of a well-posed control problem: All available measurements of plant outputs are subject to some disturbance (throughw2→v) and all control signals are penalized (throughu→z2).
4.3.1 H
∞Controller Synthesis
Theorem 4.1 (Gahinet & Apkarian [1994, Theorem 4.1] in consideration of the spe-cial structure (4.19)). Let Gbe an open-loop generalized plant with the special struc-ture(4.19). There exists a linear controller
K:
(x˙ =A x+B v
u=C x+D v (4.20)
that internally stabilizes the closed-loop interconnectionGCLsuch that kGCLk< γ if and only if
γ >max(σmax(D111•), σmax(D11•1T )) and there existX∞0 andY∞0 such that
X∞ I I Y∞
≺0 (4.21a)
ΛX∞−γ B2B2T X∞C11T B1−B2D112•
? −γ I D111•
? ? −γ I
≺0 (4.21b)
ΛY∞−γ C2TC2 Y∞B11 C1T−C2TD11•2T
? −γ I DT11•1
? ? −γ I
≺0 (4.21c)
where? denotes symmetric completion and ΛX∞ :=X∞ (A−B2C12)T + (A−B2C12)X∞,
ΛY∞ :=Y∞(A−B12C2) + (A−B12C2)T Y∞.
Proof. The proof is given in detail by Gahinet & Apkarian [1994, Theorem 4.1] and is based upon showing that feasible solutions to (4.21) imply the existence of a matrix that satisfies the bounded-real lemma for the closed-loop system. The simplified form given here immediately follows from making use of the fact thatD12 andD21 have full rank and are normalized.
A controller can be constructed by closed formulae from the open-loop plant matrices, X∞,Y∞, andγ, [e. g., Glover & Doyle 1988, Doyle et al. 1989, Gahinet & Apkarian 1994, Gahinet 1996, Zhou et al. 1995, Theorem 16.4, p. 411].
Matlab’s Robust Control Toolbox [Balas et al. 2014] provides the hinfsyn routine to solve the H∞ controller synthesis problem. This is done either based on the LMI characterization of Theorem 4.1 and convex optimization or based on a formulation by Glover & Doyle [1988] and Doyle et al. [1989] which uses Riccati equations. In the latter case, an iteration of the value for γ through a bisection algorithm is performed until a solution close enough to the optimum is found. The Riccati solution appears to be numerically more benign in most cases and very recent results by Glover & Packard [2017]
are likely to further improve reliability. The routinehinfsynalso automatically performs all necessary transformations to achieve the special form (4.19).10
10The particular structure of the McFarlane-Glover controller can further be exploited to avoid the iteration and immediately obtain a suboptimal controller as implemented in the Matlab routinesncfsyn andloopsyn.
4.3.2 LPV Controller Synthesis
A synthesis condition for gridded LPV systems was first published by Wu et al. [1995, 1996]
and is detailed by Wu [1995, Cha. 4]. Essentially identical results were also obtained around the same time by Wood [1995, Cha. 8]. The synthesis conditions can be seen as immediate extensions of theH∞ controller synthesis conditions of Theorem 4.1.
Theorem 4.2. [Wu 1995, Theorem 4.3.2, p. 81] Let Gρ be an open-loop generalized LPV plant with the special structure (4.19) defined on the domainT. There exists an LPV controller
Kρ:
(x˙ =A(ρ,ρ)˙ x+B(ρ)v
u= C(ρ)x+D(ρ)v (4.22)
that internally stabilizes the closed-loop interconnectionGCL,ρand guaranteeskGCL,ρk< γ if
γ >max
p∈Pmax σmax(D111•(p)), σmax D11•1T (p)
and if there exist symmetric positive definite matrix functions X: P 7→Rnx×nx and Y:P 7→Rnx×nx such that for all(p, q)∈ P × Q
where ?denotes symmetric completion and
ΛX(p, q) :=X(p) (A(p)−B2(p)C12(p))T+ (A(p)−B2(p)C12(p))X(p)− Proof. The proof is provided in extensive length by Wu [1995, Sec. 4.3] and is based upon showing that feasible solutions to the LMIs (4.23) imply the existence of a matrix function that satisfies Theorem 2.2 for the closed-loop system.
The parameter rate q appears affinely in the terms ΛX and ΛY and hence the con-straints (4.23) need to be satisfied for the vertices of the setQ, i. e., both for minimum and maximum values of the rates at each grid point. In order to arrive at a tractable formulation, the positive definite matrix functionsX:P 7→Rnx×nx and Y:P 7→Rnx×nx must further be formulated in terms of a predefined set of basis functions as
X(ρ) = Xa
i=1
fi(ρ)Xi, Xi∈Rnx×nx and Y(ρ) = Xb
i=1
gi(ρ)Yi, Yi∈Rnx×nx. (4.24) It is important to emphasize that the conditions used in Theorem 4.2 are only sufficient.
The first reason for this is the restriction to a quadratic-in-the-states storage function as a certificate to bound the inducedL2-norm in Theorem 2.2, which forms the basis for Theorem 4.2. In the context of LMIs, this restriction is inevitable. The second reason is related to the conservatism that is introduced by additionally restricting the search space for the parameter-dependent matricesX(ρ) and Y(ρ), which implicitly constitute such a storage function. Thus, selecting more basis functions forX(p) and Y(p) to enhance the search space usually increases performance. A controller can be constructed by closed formulae from the open-loop plant matrices and the feasible values ofX, Y, and γ as described by Wu [1995, p. 82] or Lee [1997, Theorem 4.2.5, pp. 47]. The relationship to the conditions of Theorem 4.1 is established for LTI systems and constant matrices X =1/γX∞,Y =1/γY∞ through multiplication of all inequalities (4.23) byγ.
The freely available LPVTools toolbox [Balas et al. 2015, Hjartarson et al. 2015] for Matlab provides the routine lpvsyn that implements the LPV synthesis conditions as a convex optimization problem to minimize γ. Further, basis functions as stated in Equation (4.24) can be conveniently defined and an interface with many Robust Control toolbox and Control Systems toolbox functions is provided. The routine also performs all necessary transformations to achieve the special form (4.19).11
4.3.3 Suboptimal Synthesis and Implementation
Both inH∞ and LPV control, suboptimal controllers are practically more relevant than actual “optimal” ones. Particular insight into the problem is again obtained by the McFarlane-Glover method that permits an analytical solution which, however, turns out to be singular [McFarlane & Glover 1992]. The same limiting behavior can be observed for most synthesis conditions: Being a convex optimization problem, it is not surprising that the minimum often is located on the boundary of the constraints. Hence, nearly singular matricesX andY are often obtained and as the controller reconstruction usually involves inverses of these matrices, numerical problems are commonly encountered.
Further, the optimal solution with its flat frequency response characteristic is not necessarily desirable for control problems [cf. Zhou et al. 1995, Sec. 16.9]. Specifically, nearly optimal solutions often result in unnecessarily fast controller dynamics, which can severely complicate implementation. Several remedies have been proposed, e. g., to
11Again, the special structure of the McFarlane-Glover design problem permits a simplified solution of the LPV controller synthesis problem as described by Wood [1995, Sec. 8.4–8.5].
incorporate additional (pole region) constraints [Lee 1997, Sec. 4.2.3]. The so far most successful approach, however, appears to be a relaxation of the problem after the achievable performance index has been determined. In a second step, a new synthesis problem is solved with a fixed suboptimal performance index that is 5–20 % above the optimalγ.
This often eliminates fast controller dynamics, while the controllers’ frequency response is usually indistuingishable in the relevant frequency range and almost identical results in time domain are achieved. It is further possible to use this second step to explicitly improve conditioning of the matricesX andY by solving additional optimization problems [e. g., Saupe 2013, Sec. 3.4]. A very similar approach is also implemented inlpvsyn[Balas et al. 2015].
4.3.4 Discretization
Naturally, controllers need to be implemented on a digital computer, i. e., in discrete time. It is in principle possible to use synthesis techniques that directly yield discrete time controllers [e. g., Packard 1994, Apkarian & Gahinet 1995]. The question of whether the design should be carried out in discrete time rather than continuous time therefore naturally arises. A discrete time design would automatically incorporate the delay due to the zero-order hold operation. This delay would, however, usually only be a small part of the overall delay. The remaining part, caused by sensor and actuator components as well as computation units, would still require a model. While the advantages of a discrete-time design thus seem to be limited, insight into the problem would be lost to a certain extent as engineering intuition is commonly higher developed in the continuous-time domain [cf. Apkarian et al. 1996, Sec. 12]. It therefore appears preferable to design a controller in continuous time and then to discretize the resulting controller using any available discretization scheme as discussed in detail by Apkarian et al. [1996, Sec. 12]. In particular, the standard Tustin discretization, also known as trapezoidal discretization or bilinear transformation, is applicable to LPV systems and can be formulated as [ibid., Theorem. 12.2.1, p. 205]
In Equation (4.25),krefers to the time step andT is the sampling time which has to be chosen sufficiently small. Thus,xk refers tox(t)|t=k T. This form is easy to implement in a lookup table representation, completely analog to the continuous-time state space representation.