Analysis and Control of Linear Systems - Chapter 13 pptx

Chapter 13 Methodology of the State Approach Control Designing the “autopilot” of a multivariable process, be it quasi-linear, represents a delicate thing. If the theoretical and algorithmic tools concerning the analysis and control of multivariable linear systems have largely progressed during the last 40 years, designing a control law is left to the specialist. The best engineer still has difficulties in applying his knowledge related to multivariable control acquired during his automation course. It is not a mater here to question the interest and importance of automation in the curriculum of an engineer but to stress the importance of “methodology”. The teaching of a “control methodology”, coherently reuniting the various fundamental automation concepts, is the sine qua non condition of a fertile transfer of knowledge from laboratories toward industry. The methodological challenge has been underestimated for a long time. How else can we explain the little research effort in this field? It is, however, important to underline among others (and in France) the efforts of de Larminat [LAR 93], Bourlès [BOU 92], Duke [DUC 99], Bergeon [PRE 95] or Magni [MAG 87] pertaining to multivariable control methodology. This chapter deals with a state-based control methodology which is largely inspired by the “standard state control” suggested by de Larminat [LAR 00]. Chapter written by Philippe CHEVREL. 400 Analysis and Control of Linear Systems 13.1. Introduction Controlling a process means using the methods available for it in order to adjust its behavior to what is needed. The control applied in time uses information (provided by the sensors) concerning the state of the process to react to any unforeseen evolution. Designing even a little sophisticated control law requires the data of a behavior model of the process but also relevant information on its environment. Which types of disturbances are likely to move the trajectory of the process away from the desired trajectory and which is the information available a priori on the desired trajectory? Finally, a method of designing control laws must make it possible to arbitrate among various requirements: – dynamic performances (which must be even better when the transitional variances between the magnitudes to be controlled and the related settings are weak); – static performances (which must be even better when the established variances between the magnitudes to be controlled and the related settings are weak); – weak stress on the control, low sensitivity to measurement noises (to prevent a premature wear and the saturation of the actuators, but to also limit the necessary energy and thus the associated cost); – robustness (qualitatively invariant preceding properties despite the model errors). Although this last requirement is not intrinsic (it depends on the model retained for the design), it deserves nevertheless to be discussed. It translates the following important fact. Since the control law is inferred from models whose validity is limited (certain parameters are not well known, idealization by preoccupation with simplicity), it will have to be robust in the sense that the good properties of control (in term of performances and stress on the control) apply to the process as well as to the model and this despite behavior variations. This need for arbitrating between various control requirements leads to two types of reflection. It is utopian to suppose that detailed specifications of these requirements can be formalized independently of the design approach of the control law. In practice, the designer is very often unaware of what he can expect of the process and an efficient control methodology will have as a primary role to help him become aware of the Methodology of the State Approach Control 401 attainable limits. The problem of robustness can also be considered in two ways 1 . In the first instance, modeling uncertainties are assumed to be quantified in the worst case and we seek to directly obtain a regulator guaranteeing the expected performances despite these uncertainties. At their origin, the H ∞ control [FRA 87] and the µ-synthesis [DOY 82, SAF 82, ZHO 96] pursued this goal. A more realistic version consists of preferring a two-time approach alternating the synthesis of a corrector and the analysis of the properties which it provides to the controlled system. Hence, the methodology presented in this chapter will define a limited number of adjustment parameters with decoupled effects, so as to efficiently manage the various control compromises. How can the various control compromises be better negotiated than by defining a criterion formalizing the satisfaction degree of the control considered? The compromise would be obtained by optimizing this criterion after weighting each requirement. Weightings would then play the part of adjustment parameters. A priori very tempting, this approach faces the difficulties of optimizing the control objectives and the risks of an excess of weightings which may make the approach vain. It is important in this case to define a standard construction procedure of the criterion based on meta-parameters from which the weightings will be obtained. These meta-parameters will be the adjustment parameters. The methodology proposed here falls under the previously defined principles, i.e. it proceeds by minimization of the judiciously selected standard of functional calculus. When we think of optimal control, we initially think 2 of control 2 H or ∞ H . We will prefer working in Hardy’s space 2 H (see section 13.2) for the following reasons: – the criterion, expressed by means of 2 H standard ( 2 H is a Hilbert space), can break up as the sum of elementary criteria; – control 2 H has a very fertile reinterpretation in terms of LQG control which was the subject of many research works in the past whose results can be used with benefit (robustness of LQ control, principle of separation, etc.); – the principle of the “worst case” inherent to control ∞ H is not necessarily best adapted to the principle of arbitration between various requirements. In addition, and even if the algorithmic tools for the resolution of the problem of standard ∞ H  optimization operates in the state space, the philosophy of the ∞ H approach is based more on an “input-output” principle than on the concept of state. 1 In [CHE 93] we used to talk of direct methods versus iterative methods. 2 For linear stationary systems. 402 Analysis and Control of Linear Systems In fact, the biggest difficulty is not in the choice of the standard used (working in ∞ H would be possible) but in the definition of the functional calculus to minimize. This functional calculus must standardize the various control requirements and be possible to parameterize based on a reduced number of coefficients. In the context of controls 2 H or ∞ H , it is obtained from the construction of a standard control model. This model includes not only the model of the process but also information on its environment (type and direction of input of disturbances, type of settings) and on the control objectives (magnitudes to be controlled, weightings). The principle of its construction is the essence of the methodology presented in this chapter. The resolution of the optimization problem finally obtained requires to remove certain generally allowed assumptions within the framework of the optimization problem of standard 2 H . In short, the methodological principles which underline the developments of this chapter are as follows: – to concentrate on an optimization problem so as to arbitrate between the various control requirements; – to privilege an iterative approach alternating the design of a corrector starting from the adjustment of a reduced number of parameters up to the decoupled effects and the analysis of the controlled system; – to express the control law based on intermediate variables having an identified physical direction and thus to privilege the state approach and the application of the separation principle in its development. The control will be obtained from the instantaneous state of the process and its environment. This chapter is organized as follows. Section 13.2 presents the significant theoretical results relative to the 2 H control and optimization and carries out certain preliminary methodological choices. The minimal information necessary to develop a competitive control law is listed in section 13.3 before being used in section 13.4 for the construction of the standard control model. The methodological approach is summarized in this same section and precedes the conclusion. 13.2. H 2 control The traditional results pertaining to the design of regulators by 2 H optimization and certain extensions are given in this chapter. Its aim is not to be exhaustive but to introduce all the notions and concepts which will be useful to understand the methodology suggested later on. Methodology of the State Approach Control 403 13.2.1. Standards 13.2.1.1. Signal standard Let us consider the space n L 2 of the square integrable signals on [[ ∞,0 , with value in n R . We can define in this space (which is a Hilbert space) the scalar product and the standard 3 defined below: +∞ +∞ ⎛⎞ ⎜⎟ == ⎜⎟ ⎝⎠ ∫∫ 1 2 2 00 ,()(), ()() TT xy xt ytdt x xt xtdt [13.1] The Laplace transform TL() makes the Hardy space n H 2 of analytical functions )(sX in 0)( ≥sRe and of integrable square correspond to n L 2 . Parseval’s theorem makes it possible to connect the standard of a temporal signal of n L 2 to the standard of its Laplace transform in n H 2 : ωωω π +∞ −∞ ⎛⎞ ⎜⎟ == ⎜⎟ ⎝⎠ ∫ 1 2 * 22 1 (()()) 2 xX GraphXjXjd [13.2] 13.2.1.2. Standard induced on the systems Let us consider the multivariable system defined by the proper and stable (rational) transfer matrix ()Gs or alternatively by its impulse response − ⋅= ⋅ 1 () TL ()g . y u G(s) 3 Standard whose physical importance in terms of energy is obvious. 404 Analysis and Control of Linear Systems The “H 2 standard” of the input-output operator associated with this system is defined, when it exists, by: ωωω π +∞ −∞ ⎛⎞ ⎜⎟ = ⎜⎟ ⎝⎠ ∫ 1 2 * 2 1 (()()) 2 GGraphGjGjd [13.3] Let us note that m Rtu ∈)( and p Rty ∈)( respectively the input and output of the system at moment t. Let )(tR uu , )(tR yy be the autocorrelation matrices and )( ωjS uu , )( ωjS yy the associated spectral density matrices. We recall that these matrices are defined as follows. For a given u signal we have: ττ →+∞ − =+ ∫ 1 () lim ( ) () 2 T T uu T T R ut u t dt T . For a centered random u signal, whose certain stochastic characteristics (in particular its 2 order momentum) are known, )(⋅ uu R could be also defined by the equality: )]()([)( tutuER T uu τ+=τ . The two definitions are reunited in the case of a random signal having stationarity and ergodicity properties [PIC 77]. In addition we have the relation: ττ=ω ωτ +∞ ∞− ∫ deRjS j uuuu )()( . These notations enable us to give various interpretations to the 2 H standard of G . The results of Table 13.1 are easily obtained from Parseval’s equality or the theorem of interferences [PIC 77, ROU 92]. They make it possible to conclude that 2 G is also the energy of the output signal in response to a Dirac impulse or that it characterizes the capacity of the system to transmit a white noise 4 . These interpretations will be important further on. 4. Characterized by a unitary spectral density matrix. Methodology of the State Approach Control 405 Characteristic of the input signal 2 G Significance )()( tItu m δ= 222 gyG == δ ⋅ ⎧ ⋅ ⎨ = ⎩ ( ) is of zero mean () / () () uu m u u Rt I t ωω ∞ −∞ ⎡⎤ == ⎢⎥ ⎣⎦ = ∫ 22 2 () ( (0)) (()) yy yy G E y t graph R graph S j d Table 13.1. Several interpretations of ||G|| 2 13.2.1.3. The grammians’ role in the calculation of the 2 H standard Let us consider the quadruplet ppnpmnnn RDRCRBRA ×××× ∈∈∈∈ ,,, such that: − =− + 1 () ( )Gs CsI A B D [13.4] In other words, the state () n Rtx ∈ of the system Σ evolves according to: 0 () () with: (0) () () xt A B xt x x yt C D ut ⎛⎞⎛ ⎞⎛⎞ == ⎜⎟⎜ ⎟⎜⎟ ⎝⎠⎝ ⎠⎝⎠  [13.5] The partial grammians associated with this system are defined by: ττ ττ τ τ = = ∫ ∫ 0 0 () () T T t ATA c t ATA o teBBe d teCCed G G [13.6] Table 13.2 presents the results emerging from these definitions. 406 Analysis and Control of Linear Systems Input signal characteristic Significance of grammians δ =() () m ut I t , 0 0 =x ττ τ = ∫ 0 () () () t T c txx dG δ ⋅ ⎧ ⋅ ⎨ = ⎩ ( ) is of zero mean ()/ () () uu m u u Rt I t =() ( () ()) T c tExtxtG =() 0ut , = 0 (0)xx τττ = ∫ 00 0 () ( ) ( ) t TT o xtxyydG Table 13.2. Several interpretations of grammians )(t c G and )(t o G are respectively called partial grammians of controllability and observability. In fact, −1 [()] c tG is directly connected to the minimal “control energy” necessary to transfer the system from state 0)0( =x to state 1 )( xtx = [KWA 72]. Basically, τ τ − − = 1 () 1 11 () [ ( )] T At T c uBe txG 1 0 tτ <≤ is the minimal energy control [] ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ττ − ∫ 1 1 1 0 )()()( x t xuu c T t T G that changes the state )(⋅x from 0 0 =x to 0=t to 1 x to 1 tt = . There are also the following equivalences: – ),( BA is controllable 0)( >>∀⇔ t c G0,t ; – ),( AC is observable 0)( >>∀⇔ t o G0,t . It is shown without difficulty that )(t c G and )(t o G are solutions of Lyapunov differential equations: =+ +  () () () TT Gt A t tA BB ccc GG =(0) 0 c G =++  () () () TT Gt A t tACC ooo GG =(0) 0 o G [13.7] The partial grammians can be effectively calculated by integrating this system of first order differential equations (see section 13.6.1). Methodology of the State Approach Control 407 The “total” grammians (this qualifier is generally omitted) result from the partial grammians by: )(lim T c T c GG +∞→ = and )(lim T o T o GG +∞→ = . Their existence results from the stability of the system. They are the solution of Lyapunov algebraic equations obtained by canceling the derivatives )(t c G  and )(t o G  : 0=++ TT cc BBAA GG and 0=++ CCAA T oo T GG . The following important property is therefore inferred. Let () sG be the transfer matrix defined by the presumed minimal realization ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 :)( C BA sG . Then: () ==GG 2 2 () () TT oc Gs GraphB B GraphC C [13.8] Numerically, standard 2 H of ()Gs could be obtained by resolution of an Lyapunov algebraic equation obtained from the state matrices CBA ,, . Let us note that matrix ()Gs must be strictly proper for the existence of 2 ()Gs . A last interesting interpretation of standard 2 H of ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 :)( C BA sG is as follows. Let m BBB ••• ,, 21 be the columns of B . Let Li y be the free response of the system on the basis of the initial condition ii Bx • = 0 . It is verified then that the following identity is true: =++L 222 2 12 222 2 () LL Lm Gs y y y [13.9] Thus, standard 2 H gives, for a system whose state vector consists of internal variables easy to interpret, an energy indication on its free response for a set of initial conditions contained in )Im(B . 13.2.2. H 2 optimization 13.2.2.1. Definition of the standard H 2 problem [DOY 89] Any closed loop control can be formulated in the standard form of Figure 13.1. 408 Analysis and Control of Linear Systems Figure 13.1. Standard feedback diagram The quadripole G , also called a standard model, and feedback K are supposed to be defined as follows, by using the transfer matrices ()Gs and ()Ks and their realization in the state space: ⎡ ⎤⎛⎞⎡ ⎤⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎢⎥⎢⎥ ==⇔= ⎜⎟ ⎜⎟ ⎜⎟ ⎢⎥⎢⎥ ⎝⎠ ⎜⎟ ⎜⎟ ⎢⎥⎢⎥ ⎣ ⎦⎝⎠⎣ ⎦⎝⎠  12 12 11 12 1 11 12 1 11 12 21 22 2 21 22 2 21 22 () () () : () () A BB x ABBx Gs Gs Gs C D D z C D D w GsGs CD D y CD D u − =+ − 1 () ( ) KK K K Ks D C sI A B [13.10] NOTE 13.1.– the size of each matrix results from the size of the various signals: 2121 ,,,, ppnmm RyRzRxRuRw ∈∈∈∈∈ . The closed loop system of input w and output z, noted by zw T , is obtained: () () ()( ) () () () () ()()() sGsKsGIsKsGsGsKsGFsT l 21221211zw 1 , − ∆ −+== − =+ − 1 () bf bf bf bf D CsIA B [13.11] [...]... ⎥ ⎥ D22 ⎥ ⎦ [13. 19] 418 Analysis and Control of Linear Systems 0 We will assume D11 = DWs D11 DWe = 0 By construction, the modes of Ws (s) are unobservable by y, whereas the modes of We (s) are non-controllable by u If these modes are unstable, the standard model G(s) is non-stabilizable by u and nondetectable by y The standard H 2 problem cannot be solved (we are outside its context of hypothesis)... properties of robustness (see Chapter 6 and [SAF 77]) The exteriority of the Nyquist place with respect to the Kalman circle guarantees good gain and phase margins, as well as good robustness with respect to static non-linearities (criterion of the circle [SAF 80]) and a certain type of dynamic uncertainties5 These properties are obtained at the beginning of the process Figure 13. 4 Analysis of robustness of. .. the standard H2 problem ( A, B2 ) stabilizable and (C2, A) detectable ⎛ A − jωI ∀ω ∈ R, ⎜ ⎜ C 1 ⎝ B2 ⎞ ⎟ and D12 are of full rank per column D12 ⎟ ⎠ ⎛ A − jωI B1 ⎞ ⎟ and D21 are of full rank per row These hypotheses ∀ω ∈ R, ⎜ ⎜ C D21 ⎟ 2 ⎝ ⎠ are easily understood if it is known that at optimum, the poles of Tzw (s) tend toward 412 Analysis and Control of Linear Systems the zeros of transmission of G12... standard discrete problem for which K dH2 (z) would be the solution 9 B0 and ET represent respectively the 0 order blocker and the sampling operator in accordance with Chapter 3 424 Analysis and Control of Linear Systems Figure 13. 7 Discretization of the standard H 2 problem Discretizing problem H 2 does not imply discretizing the standard model by writing Gd = ET G B0 By doing this, the behavior of. .. operation environment of the system 428 Analysis and Control of Linear Systems 13. 3.2 Modeling the environment of the system to adjust Controlling any system consists of controlling its actuators so that it can fulfill the required function The control law naturally depends on the function to be accomplished and on the conditions under which it must be accomplished The magnitudes to be controlled were already... case of a standard model that does not have unstable modes unobservable by y This restrictive and simplifying hypothesis will not block the “State Standard Control type methodological developments From the H 2 generalized problem, we can establish the following result which shows the presence of an internal model [WON 85] within the regulator 422 Analysis and Control of Linear Systems Theorem 13. 3... hypotheses of evolution noise w x and measurement noise w y previously defined Methodology of the State Approach Control 413 Figure 13. 2 Standard form for LQG control Figure 13. 3 LQG structure (state feedback/observer) The resulting control law illustrated in Figure 13. 3 has the structure of the state feedback/observer: ˆ ⎧ u = −K LQ x ⎪ ⎨ ˆ ˆ ⎪ x = ( A − LFK C2 )x + B2u + LFK ( y − C2 x ) ⎩ ˆ [13. 17]... dynamic and stochastic properties as the discretized state x(kTe ) of G H 2 It is 426 Analysis and Control of Linear Systems shown then that ∆ z d 2 = ∑ z dk T z dk = z 2 Finally, k y dk is obtained by discretization of the white noise D21w NOTE 13. 4.– we have, by definition of the H 2 standard in discrete time, 1 +π ⎛ ⎞ 2 Te ⎜ 1 ⎟ Tzd wd = ⎜ Trace (Tz∗ wd (e jωTe )Tz w (e jωTe )) dω ⎟ Standard... solution P3 of the Riccati equation reduced to the controllable part by u is solution of 1 ⎛ Σ Σ =⎜ 1 ⎜S Σ ⎝ a 1 Σ1 S a T ⎞ ⎟ S a Σ1 S a T ⎟ ⎠ where solution Σ1 of the Riccati equation reduced to the observable part by y is solution of 3 Let us give the idea of the equivalence proof of properties 1 and 2., the equivalence of properties 3 and 4 resulting by duality Methodology of the State Approach Control. .. , we build the standard model GH 2 (s ) problem ⎞ ⎛ 12 ⎜ Ro 0⎟ ⎟ ⎜ 0 ⎟ GMCC (s ) ⎜ 0 ⎟ ⎜ I⎟ ⎜ 0 ⎟ ⎜ ⎠ ⎝ (see 0 1 Qo 2 0 ⎞ 0⎟ ⎛ A ⎟ ⎜ 0 ⎟ := ⎜ C1 ⎟ I ⎟ ⎜ C2 ⎟ ⎝ ⎠ Figure B1 0 D21 B2 ⎞ ⎟ D12 ⎟ 0 ⎟ ⎠ 13. 12): 434 Analysis and Control of Linear Systems Figure 13. 12 Definition of the H2 problem The H 2 problem that consists of calculating K (s) ensuring the internal stability of GSAR (s) and minimizing Fl . Definition of the standard H 2 problem [DOY 89] Any closed loop control can be formulated in the standard form of Figure 13. 1. 408 Analysis and Control of Linear Systems Figure 13. 1. Standard feedback. the poles of )(sT zw tend toward 412 Analysis and Control of Linear Systems the zeros of transmission of )( 12 sG and )( 21 sG . In addition, the remaining invariant zero are non-controllable. talk of direct methods versus iterative methods. 2 For linear stationary systems. 402 Analysis and Control of Linear Systems In fact, the biggest difficulty is not in the choice of the standard