Robust Control Toolbox User's Guide

Optional System Data Structurecommand desg mksys(ag,bg,cg,dg,eg,'des'); packs a descriptor systeminto desg, etc.Table 1-1:TypeV1, V2, V3, , VnDescription'ss'(a,b,c,d,ty)Standard State-Space(default)'des'(a,b,c,d,e,ty)Descriptor -Port ty)Two-Port Descriptor'gssv(sm,dimx,dimu,dimy,ty)General State-Space'gdes'(e,sim,dimx,dimu,dimy,ty)General Descriptor'gpsm'(psm,deg,dimx,dimu,dimy,ty)General PolynomialSystem Matrix'tf'(num,den,ty)Transfer Function'tfm'(num,den,m,n,ty)Transfer Function Matrix'imp'(y,ts,nu,ny)Impulse Responsebranch: This function recovers the matrices packed in a system or tree variableselectively. For example,[D11,C2] branch(TSS,'d11,c2');recovers the matrices D11 and C2 from the system TSS andag branch(ssg,'a');recovers the matrix ag from the state-space system ssg.To recover all the matrices from ssg at once, you may type[ag,bg,cg,dg] branch(ssg);1-5

1Tutorialtree: This function provides a general tool for creating hierarchical datastructures containing matrices, strings and even other trees. It is used as asubroutine by mksys. For example, if you wish to keep track of the two-portplant (A,B1,B2,.), along with the controller (af,bf,cf,df), the frequencyresponse [w;sv] along with the name Aircraft Design Data, you simply do thefollowingfr tree('w,sv',w,sv);DesignData rcraft Design Data');Figure 1-1, Branch Structure of the tree Variable shows the branch structureof the tree variable DesignData. This tree variable has two levels, since thebranches named plant, controller, and freq are themselves trees. However,there is in general no limit to the number of levels a tree can have.To recover the variable name from the first level of the tree DesignData, wetypename branch(DesignData,'name')ans Aircraft Design DataThe list of names of “root branches” of the tree is always stored in the tree asbranch 0. For example, to find the names of the root branches in the treevariable DesignData, type the followingbranch(DesignData,0)ans plant,controller,freq,nameTo recover the value of the matrix c1 in the branch plant of the second level ofthe tree DesignData, we typeC1 branch(DesignData,'plant/c1');1-6

Optional System Data eqAircraftDesignDatanameDesignDataFigure 1-1: Branch Structure of the tree VariableThe M-files in the Robust Control Toolbox have been reinforced to optionallyaccept the system data structure to simplify user interaction and to reduce theamount of typing required. Whenever a Robust Control Toolbox functionencounters a tree variable representing a system among it input arguments, itautomatically checks to see if the variable is in fact a system. If it is a system,then the function automatically expands the input argument list, replacing thesystem variable by the matrices stored in it. For example, the following twocommands perform the same 21,D22);The latter, longer form illustrates the fact that the use of system variables isentirely optional in the Robust Control Toolbox. The traditional butcumbersome method of passing system matrices one at a time as multiplearguments to a function is still acceptable, thus ensuring compatibility withother MATLAB toolboxes and earlier versions of the Robust Control Toolbox.See the Reference chapter for details on mksys, branch, and tree.1-7

1Tutorial Singular Values, H2 and H Normsm nThe singular values of a rank r matrix A C, denoted σ i are thenon-negative square-roots of A * A the eigenvalues of ordered such thatσ 1 σ 2 σ p , p min { m , n }If r p then there are p – r zero singular values, i.e.,σr 1 σr 2 σp 0There exist two unitary matrices U Cm nmatrix Σ Rsuch thatA U ΣV UΣr 0m m, V Cm nand a diagonalV 0 0where Σ r diag ( σ 1 ,σ 2 , ,σ r ) ; this is called the singular-value decomposition(SVD) of matrix A. The greatest singular value σ 1 is sometimes denotedσ ( A ) σ1If A is a square n n matrix, then the n-th singular value (i.e., the leastsingular value) is denoted σ ( A ) σnSome useful properties of singular values areAx1 σ ( A ) max x C n ----------x2Axσ ( A ) min x C n -----------x3 σ ( A ) λ i ( A ) σ ( A ) , where λ i denotes the i-th eigenvalue of A.1-8

Singular Values, H2 and H Norms4 If A5 If A–11exists, σ ( A ) -----------------–1σ(A )–11exists, σ ( A ) -----------------–1σ(A )6 σ ( αA ) α σ ( A )7 σ(A B) σ(A) σ(B )8 σ ( AB ) σ ( A )σ ( B )9 σ(A) – σ(E ) σ(A E ) σ( A) σ(E )10 max { σ ( A ) ,σ ( B ) } σ ( [ AB ] ) 2 max { σ ( A ) ,σ ( B ) }11 max i ,j a i ,j σ ( A ) n max i ,j a i ,jn12σ i2 i 1 Trace ( A * A )Property 1 is especially important because it establishes the greatest singularvalue of a matrix A as the maximal “gain” of the matrix as the input vector “x”varies over all possible directions.m n, p min { m ,n } , defineFor stable Laplace transform matrices G ( s ) Cthe H2-norm and the H -norm terms of the frequency-dependent singularvalues of G ( jω ) :H2-norm: G 2 p – ( σi ( G ( jω ) ) )2dω1--2i 1 H -norm: G supσ ( G ( jω ) )ω( sup: the least upper bound )1-9

1TutorialThe Robust Control ProblemIn the past two decades there have been great advances in the theory for thedesign of robustly uncertainty-tolerant multivariable feedback control systems[8, 9]. Many of the questions that created the much lamented “gap” of the1970’s between the theory and practice of control design have been resolved, atleast partially, in the wake of the renewed concern of control theorists withsuch feedback issues such as stability margin, sensitivity, disturbanceattenuation and so forth. Out of this renewed concern has emerged the singularvalue Bode plot as a key indicator of multivariable feedback systemperformance (e.g., [12, 29]). The singular value thus joins such previously usedmeasures of multivariable feedback system performance as dominant polelocations (related to disturbance rejection bandwidth and transient response),transmission zeros (related to steady-state response and “internal models”)and rms error of control signals (from the L2 Wiener-Hopf/LQG optimal controltheory, [1, 45, 46]).The real problem in robust multivariable feedback control system design is tosynthesize a control law which maintains system response and error signals towithin prespecified tolerances despite the effects of uncertainty on the system.Uncertainty may take many forms but among the most significant are noise/disturbance signals and transfer function modeling errors. Another source ofuncertainty is unmodeled nonlinear distortion. Uncertainty in any form is nodoubt the major issue in most control system designs. Consequently peoplehave adopted a standard quantitative measure for the size of the uncertainty,viz., the H norm.The general robust control problem is described mathematically as follows (SeeFigure 1-2, Canonical Robust Control Problem):Given a multivariable plant P(s), find a stabilizing controller F(s) such that theclosed-loop transfer function T y 1 u 1 satisfies1-------------------------------------- 1K M(T y u ( jω ))1 1wheredefKM ( Ty u ) 1 11-10inf { σ ( ) det ( ( I – T y u ) ) 0 }1 1

The Robust Control Problemwith diag ( 1 , , n 1P22PLANTy2u12F(s)CONTROLLERFigure 1-2: Canonical Robust Control ProblemThe condition 1 K M(T y u ( jω )) 1 is “robustness criterion.” The quantity K M1 1is “size” of the smallest uncertainty , as measured by the singular value ateach frequency, that can destabilize the closed-loop system. The function K Mis the so-called diagonally perturbed multivariable stability margin (MSM)introduced by Safonov and Athans [30, 32], the reciprocal of which is known as1µ, the structured singular value (SSV) [13] —. i.e., K M --- . More precisely,µwhen n is not present, this problem is called the robust stability problem.Doyle, Wall and Stein [14] introduced the extra uncertainty n to representthe performance specification σ ( T ed ( jω ) ) 1 which, according to their robustperformance theorem, is satisfied if and only if 1 K M(T y 1 u 1 ( jω )) 1 . Thus, theproblem set-up in Figure 1-2, Canonical Robust Control Problem completelyaddresses the issues in robust control system design, i.e., robustness andperformance.1-11

1TutorialUnfortunately the computation of K M(T y1 u 1) involves a nonconvexoptimization over and so cannot, in general, be solved by the standardgradient-descent nonlinear programming techniques for which convexity ofconstraints and cost is required to assure convergence. Fortunately,computable upper bounds on 1 K M do exist and have provided simplealternatives for computing K M :–11-------------------------- µ(T y 1 u 1) inf DT y 1 u 1 D D p T y 1 u 1 Dp– 1 K m(T y u )D D11where D p D denotes the Perron optimal scaling matrix [32], and D : { diag ( d 1 I , ,d n I ) d i 0 } . Clearly, T y1 u 1is also an upper bound on 1 K M , albeit possibly a very conservative one. If any of the upper boundssatisfies the robust performance constraints, it is sufficient to guarantee thatµ, or K M , satisfies them as well.Therefore, from a robust control synthesis point of view, the problem is to finda stabilizing F(s) to “shape” the µ(T y 1 u 1) function (or its upper bounds) in thefrequency domain. On the other hand, from a robust control analysis point ofview, the problem is to compute the MSM K m ( T y 1 u 1 ) , (or its bounds).Structured and Unstructured UncertaintyPractically, each of the i ’s (i 1, , n) may itself be a matrix and represent adifferent kind of physical uncertainty. Two types of uncertainty are defined inrobust control — unstructured and structured.Unstructured uncertainty usually represents frequency-dependent elementssuch as actuator saturations and unmodeled structural modes in the highfrequency range or plant disturbances in the low frequency range. Theirrelations to the nominal plant can be either additiveG G Aor multiplicativeG ( I M )GBoth can be considered as norm bounded quantities, i.e., using H norm r. where r is a given positive number.1-12

The Robust Control ProblemFigure 1-3, Additive and Multiplicative Unstructured Uncertainty shows theblock diagrams of these two unstructured uncertainties.TRUE PLANT GA FG-M(s)TRUE PLANT GM F G-M(s)Figure 1-3: Additive and Multiplicative Unstructured UncertaintyStructured Uncertainty represents parametric variations in the plantdynamics, for example:1 Uncertainties in certain entries of state-space matrices (A, B, C, D), e.g., theuncertain variations in an aircraft’s stability and control derivatives.2 Uncertainties in specific poles and/or zeros of the plant transfer function.3 Uncertainties in specific loop gains/phases.1-13

1TutorialThe very general setup in Figure 1-2, Canonical Robust Control Problemallows a control system designer to capture all these uncertainties, bothstructured and unstructured, and formulate them into the design. The providessoftware tools for robustness analysis and robust control law synthesis withinthis very general framework.Positive Real and Sector UncertaintyThe setup of the robust control problem in Figure 1-2, Canonical RobustControl Problem handles much more than just the case of i ( jω ) satisfying i 1 . Using the sector transform [50, 28], this setup readily extends toadmit transfer function matrix i (s)’s and even nonlinear i ’s satisfying ageneral, possibly frequency-dependent sector condition.Definition: Given matrices A(s) and B(s) and let (s) be a stable transferfunction matrix. IfRe [ ( y – Ax ) ( y – Bx ) ] 0for all s j ω and y ( jω )x , then we say ( s ) sector[ A ,B ]More generally, if A11(s), S12(s), S21(s), S22(s) are stable transfer functionmatrices and ifRe [ ( S 11 ( s )x S 12 ( s )y ) ( S 21 ( s )x S 22 ( s )y ) ] 0for all s j ω and all y ( jω )x , then we say ( s ) sector [ S ( s ) ]For example, physically-dissipative force-velocity transfer function matricessuch as those associated with mechanical structures having collocatedactuators and sensors are positive real, i.e., inside sector[0, ], and thetransformationy y uu y u1-14

The Robust Control Problemtransforms a positive-real relation u y into an equivalent relation ũ ỹsatisfying 1The case of general A(s), B(s) matrices may be handled similarly.The function sectf.m in the Robust Control Toolbox allows you to perform thesector transform in the state-space framework. See the Reference section fordetails.Robust Control AnalysisThe goal of robust analysis is to measure the Multivariable Stability Margin(MSM) “seen” by the uncertainties using a proper, nonconservative analyticaltool. In other words, we are interested in finding out how big can be beforeinstability occurs.Two tasks are involved in computing the MSM:Task 1: Define the uncertainty modelTask 2: Pull out the uncertainty channels (structured or unstructured) into aM- form as shown in Figure 1-4, Robust Analysis M- Diagram.121.SYSTEM WITH ACTUATORS,SENSORS, CONTROLLER,.M(s).34Figure 1-4: Robust Analysis M- Diagram1-15

1TutorialFollowing are examples of modeling different types of uncertainties in the M- block diagram form.Example 1: Modeling Unstructured Uncertainty. The following plant transfer functionthat represents a spacecraft’s rigid body dynamics and one boom structuralmode (see the Case Studies section for more details).1G ( s ) -------------------------2 2s (s 2)1If the nominal model is G ----- , then (see Figure 1-5, Bode Plots of Additives22–1s 1– A G – G -------------------------- ; M ( s ) – F ( I FG )2 2s (s 2)and Multiplicative Uncertainty)G(s)10050DBDB0-100-20010 -210 -110 010 1Rad/Sec-10010 -210 2Additive Unc.6010 -110 010 1Rad/Sec10 2Multiplicative Unc.40DB50DB0-501000-50-10010 -2Gbar(s)100200-2010 -110 010 1Rad/Sec10 2-4010 -210 -110 010 1Rad/Sec10 2Figure 1-5: Bode Plots of Additive and Multiplicative Uncertainty1-16

The Robust Control Problem– M ( G – G ) G–12s 1 --------------- ;2s 2– M ( s ) G F ( I GF )–1Example 2. Modeling Structured Uncertainty . This example shows how to pull outstructured uncertainties from state-space A and B matrices.The state-space model of the lateral-directional dynamics of a fighter aircraftis shown below [5].L′ δ L′ δL′ p L′ r L′ β V T parp·δar· N′ p N′ r N′ β V T r N′ δ a N′ δ rδrv·Yp Yr Yβ VT vYδa Yδrp0 1 VT180 0 --------r·π cos α sin α 0µ appvβwhere the “primed” derivatives with respect to ( p ,r ,β ,δ a ,δ r ) are defined asI xx I zzI xzL′ L ------- N ---------------------------------- I xx2I xx I zz – I xzI xx I zzI xzN′ N ------- L ---------------------------------- I xx I I – I 2xx zzxzThe aircraft is trimmed at degrees angle of attack, flying at sea level with atotal velocity V T of 334.9 ft/sec. The states to be stabilized are body-axis rollrate (p), yaw rate (r) and the velocity component along the y-axis (v). Thevariables to be controlled (tracked) are the roll-rate about the velocity vector·( µ ) and the sideslip angle (β). The control actuators whose dynamics areignored in this analysis are aileron (δa)and rudder (δr).1-17

1TutorialThe plant data is– 1 9953– 1 0093A B1: 39 8500C1 D1-----------------------056 89270 7513– 0 1518– 331 90-----------------------06 7840– 0.02990.0060– 0.1673--------------------– 0.171100.0906– 0.00240.0204-----------------000.0298– 0.02040.2284-----------------00The state-space set-up for the robustness evaluation can be formulated usingthe block diagram in Figure 1-6, Pulling Out Parametric Uncertainties.B2 D2C2D1U B1 C1 AFigure 1-6: Pulling Out Parametric Uncertainties1-18 Y

The Robust Control ProblemThe equations associated with the block diagram are·x Ax B 1 u 1 B 2 y 2y1 C1 x D1 u1y 2 C 2 x D 2 u 1which lead to the perturbed state-space system ·x ( A B 2 C 2 )x ( B 1 B 2 D 2 ) u 1 A Bwhere matrices B2, C2 and D2 play the roles of putting the parametricuncertainty block into a diagonal structure.If L′ p ,N′ p ,L′ r ,N′ r ,L′ β ,N′ β and L′ δ ,L′ δ ,N′ δ N′ δ are the perturbations forararthe A and B1 matrices respectively, then the associated , B2 and C2 will havethe following structure diag ( L′p , N′p , L′r , N′ r , L′β , N′β , L′ δ , N′ δ , L′δ , N′ δ )aarr1 0 1 0 1 0 1 0 1 0B2 0 1 0 1 0 1 0 1 0 10 0 0 0 0 0 0 0 0 0T1 1 0 0 0 0 0 0 0 0C2 0 0 1 1 0 0 0 0 0 00 0 0 0 1 1 0 0 0 0TD2 0 0 0 0 0 0 1 1 0 00 0 0 0 0 0 0 0 1 11-19

1TutorialThe overall augmented plant becomesAB1 B2P ( s ) C1 D1 0C2 D2 0The linear fractional transformation lftf can be used to close the controllerfeedback loop F(s) around the plant from u1 to y1. Then, the transfer functionM(s) “seen” by the uncertainty blocks is the transfer function from u2 to y2.Example 3. Modeling Nonlinear Uncertainty. A saturation nonlinear element can bemodeled as unstructured uncertainty sector bounded element insidesector[0,1]which, according to the nonlinear stability results of Sandberg andZames [26, 50], may be effectively modeled as an uncertain lineartime-invariant element whose Nyquist locus lies inside a complex disk ofradius 0.5 centered on the real axis at 0.5 (See Figure 1-7, ModelingNonlinearity as Unstructured Uncertainty). This uncertain lineartime-invariant element may thus be decomposed as 0.5 A where the1-20

The Robust Control Problemadditive uncertainty A is bounded by A 0.5 . For robust stabilityM 1, M 2 .1XY10.5A0.5-2-1 X1 2Y-0.5Infinity Norm 0.5Figure 1-7: Modeling Nonlinearity as Unstructured UncertaintyRobust Analysis — Classical ApproachFirst, let’s recall classical definitions of SISO stability margin (robustness).Consider the following block diagram (Figure 1-8, Classical Gain/PhaseMargins). The gain margin can be defined as the variation of real( ), and thephase margin can be defined as the variation of imag( ). On the Nyquist plotthey are simply the intersections of loop transfer function on unit circle (phasemargin) and real-axis (gain margin)

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