New Robust Tracking and Stabilization Methods for Significant Classes of Uncertain Linear and Nonlinear Systems

There exist many mechanical, electrical, electro-mechanical, thermic, chemical, biological and medical linear and nonlinear systems, subject to parametric uncertainties and non standard disturbances, which need to be efficiently controlled. Indeed, e.g. consider the numerous manufacturing systems (in particular the robotic and transport systems,...) and the more pressing requirements and control specifications in an ever more dynamic society. Despite numerous scientific papers available in literature (Porter and Power, 1970)-(Sastry, 1999), some of which also very recent (Paarmann, 2001)-(Siciliano and Khatib, 2009), the following practical limitations remain: 1. the considered classes of systems are often with little relevant interest to engineers; 2. the considered signals (references, disturbances,...) are almost always standard (polynomial and/or sinusoidal ones); 3. the controllers are not very robust and they do not allow satisfying more than a single specification; 4. the control signals are often excessive and/or unfeasible because of the chattering. Taking into account that a very important problem is to force a process or a plant to track generic references, provided that sufficiently regular, e.g. the generally continuous piecewise linear signals, easily produced by using digital technologies, new theoretical results are needful for the scientific and engineering community in order to design control systems with non standard references and/or disturbances and/or with ever harder specifications. In the first part of this chapter, new results are stated and presented; they allow to design a controller of a SISO process, without zeros, with measurable state and with parametric uncertainties, such that the controlled system is of type one and has, for all the possible uncertain parameters, assigned minimum constant gain and maximum time constant or such that the controlled system tracks with a prefixed maximum error a generic reference with limited derivative, also when there is a generic disturbance with limited derivative, has an assigned maximum time constant and guarantees a good quality of the transient. The proposed design techniques use a feedback control scheme with an integral action (Seraj and Tarokh, 1977), (Freeman and Kokotovic, 1995) and they are based on the choice of a


Introduction
There exist many mechanical, electrical, electro-mechanical, thermic, chemical, biological and medical linear and nonlinear systems, subject to parametric uncertainties and non standard disturbances, which need to be efficiently controlled.Indeed, e.g.consider the numerous manufacturing systems (in particular the robotic and transport systems,…) and the more pressing requirements and control specifications in an ever more dynamic society.Despite numerous scientific papers available in literature (Porter andPower, 1970)-(Sastry, 1999), some of which also very recent (Paarmann, 2001)- (Siciliano and Khatib, 2009), the following practical limitations remain: 1. the considered classes of systems are often with little relevant interest to engineers; 2. the considered signals (references, disturbances,…) are almost always standard (polynomial and/or sinusoidal ones); 3. the controllers are not very robust and they do not allow satisfying more than a single specification; 4. the control signals are often excessive and/or unfeasible because of the chattering.Taking into account that a very important problem is to force a process or a plant to track generic references, provided that sufficiently regular, e.g. the generally continuous piecewise linear signals, easily produced by using digital technologies, new theoretical results are needful for the scientific and engineering community in order to design control systems with non standard references and/or disturbances and/or with ever harder specifications.
In the first part of this chapter, new results are stated and presented; they allow to design a controller of a SISO process, without zeros, with measurable state and with parametric uncertainties, such that the controlled system is of type one and has, for all the possible uncertain parameters, assigned minimum constant gain and maximum time constant or such that the controlled system tracks with a prefixed maximum error a generic reference with limited derivative, also when there is a generic disturbance with limited derivative, has an assigned maximum time constant and guarantees a good quality of the transient.The proposed design techniques use a feedback control scheme with an integral action (Seraj and Tarokh, 1977), (Freeman and Kokotovic, 1995) and they are based on the choice of a suitable set of reference poles, on a proportionality parameter of these poles and on the theory of externally positive systems (Bru and Romero-Vivò, 2009).The utility and efficiency of the proposed methods are illustrated with an attractive and significant example of position control.In the second part of the chapter it is considered the uncertain pseudo-quadratic systems of the type limited and models possible disturbances and/or particular nonlinearities of the system.For this class of systems, including articulated mechanical systems, several theorems are stated which easily allow to determine robust control laws of the PD type, with a possible partial compensation, in order to force y and y $ to go to rectangular neighbourhoods (of the origin) with prefixed areas and with prefixed time constants characterizing the convergence of the error.Clearly these results allow also designing control laws to take and hold a generic articulated system in a generic posture less than prefixed errors also in the presence of parametric uncertainties and limited disturbances.Moreover the stated theorems can be used to determine simple and robust control laws in order to force the considered class of systems to track a generic preassigned limited in "acceleration" trajectory, with preassigned majorant values of the maximum "position and/or velocity" errors and preassigned increases of the time constants characterizing the convergence of the error.

Problem formulation and preliminary results
Consider the SISO n-order system, linear, time-invariant and with uncertain parameters, described by , where ˆv K and max τ are design specifications, or 2. condition 1. is satisfied and, in addition, in the hypothesis that the initial state of the control system is null and that (0) (0) 0 rd − = , the tracking error  where the maximum variation velocity ˆrd δ − $ $ of () () rt dt − is a design specification.Remark 1. Clearly if the initial state of the control system is not null and/or (0) (0) 0 rd −≠ (and/or, more in general, () () rt dt − has discontinuities), the error () et in ( 2) must be considered unless of a "free evolution", whose practical duration can be made minus that a preassigned settling time ˆa t .Remark 2. If disturbance d does not directly act on the output y , said d y its effect on the output, in (2) d $ must be substituted with d y $ .This is one of the main and most realistic problem not suitable solved in the literature of control (Porter et al., 1970)-(Sastry, 1999).There exist several controllers able to satisfy the 1. and/or 2 specifications.In the following, for brevity, is considered the well-known state feedback control law with an integral (I) control action (Seraj and Tarokh, 1977), (Freeman and Kokotovic, 1995).By posing in the Laplace domain the considered control scheme is the one of Fig. 2. and also consider the possible uncertainty of a g .Finally, it is clear that, for the controllability of the process, the parameter b must be always not null.In the following, without loss of generality, it is supposed that 0. b − > Remark 4. In the following it will be proved that, by using the control scheme of Fig. 2, if (2) is satisfied then the overshoot of the controlled system is always null.From the control scheme of Fig. 2 from which, if all the poles of ( ) where Remark 5. Note that, while the constant gain v K allows to compute the steady-state tracking error to a ramp reference signal, v H , denoted absolute constant gain, allows to obtain t ∀ an excess estimate of the tracking error to a generic reference with derivative.On this basis, it is very interesting from a theoretical and practical point of view, to establish the conditions for which vv HK = .In order to establish the condition necessary for the equality of the absolute constant gain v H with the constant gain v K and to provide some methods to choose the poles P and ρ , the following preliminary results are necessary.
Theorem 2. Let be ,,1 , 2 , . . ., , ii i aa ai n ⎦ the nominal values of the parameters of the process and ˆPP ρ = the desired nominal poles.Then the parameters of the controller, designed by using the nominal parameters of the process and the nominal poles, turn out to be: Moreover the polynomial of the effective poles and the constant gain are: where: Proof.The proof is obtained by making standard manipulations starting from (5), from the second of (7) and from (10).For brevity it has been omitted.
Proof.The proof is obtained by making standard manipulations and for brevity it has been omitted.Now, as pre-announced, some preliminary results about the externally positive systems are stated.
Theorem 4. Connecting in series two or more SISO systems, linear, time-invariant and externally positive it is obtained another externally positive system.
Proof.If 1 () Ws and 2 () Ws are the transfer functions of two SISO externally positive systems then ( )

L
. From this and considering that ( ) the proof follows.Theorem 5. A third-order SISO linear and time-invariant system with transfer function i.e. without zeros, with a real pole p and a couple of complex poles j α ω ± , is externally positive iff p α ≤ , i.e. iff the real pole is not on the left of the couple of complex poles.
Proof.By using the translation property of the Laplace transform it is = is composed by a succession of positive and negative alternately waves.Therefore the integral () i vtof this signal is non negative iff the succession of the absolute values of the areas of the considered semi-waves is non decreasing.Clearly this fact occurs iff the factor () p t e α − is non increasing, i.e. iff 0 p α − ≤ , from which the proof derives.From Theorems 4 and 5 easily follows that: -a SISO system with a transfer function without zeros and all the poles real is externally positive; -a SISO system with a transfer function without zeros and at least a real pole not on the left of every couple of complex poles is externally positive.By using the above proposed results the following main results can be stated.

First main result
The following main result, useful to design a robust controller satisfying the required specifications 1, holds.Theorem 6. Give the process (3) with limited uncertainties, a set of reference poles P and some design values ˆv K and max τ .If it is chosen bb − = and nn aa + = then ˆK ρ ρ

∀≥
, where the constant gain v K of the control system of Fig. 2, with a controller designed by using ( 21 Proof.The proof of the first part of the theorem easily follows from ( 23) and from the fact that bb − = and nn aa + = .
In order to prove the second part of the theorem note that, from ( 22), ( 24), ( 25) and ( 26), for ˆτ ρ ρ the roots of ( ) ds # are equal to the ones of ˆ() ds and the zeros of ˆ() ns are always on the right of the roots of ˆ() ds and on the left of the imaginary axis (see Figs. 3, 4; from Fig. 4 it is possible to note that if the poles P are all in 1 − then the zeros of ˆ() ns have real part equal to ˆ/2 ρ − ), it is that the root locus of ( ) ds # has a negative real asymptote and n branches which go to the roots of ˆ() ns .From this consideration the second part of the proof follows.From Theorems 3 and 6 several algorithms to design a controller such that   A very simple algorithm is the following.Algorithm 1 Step 1.By using ( 33) and ( 34 Step 2. ρ is iteratively increased, if necessary, until by using (28) and Kharitonov's theorem, the polynomial  (Butterworth, 1930), (Paarmann, 2001).

Second main result
The following fundamental result, that is the key to design a robust controller satisfying the required specifications 2., is stated.the initial state of the control system of Fig. 2 with îi kk = is null and that (0) (0) 0 rd −= , the corresponding tracking error ()  et , always ,, ii i aa a Moreover the overshoot s is always null.
Proof.Note that the function ( ) p Ss given by ( 11) is ( ) and hence the proof.
Remark 7. The choice of P and the determination of a ρ such that (36) is valid, if the uncertainties of the process are null, are very simple.Indeed, by using Theorems 4 and 5, it is sufficient to choose P with all the poles real or with at least a real pole not on the left of each couple of complex poles (e.g.

{ }
1, 1 and then to compute ρ by using relation If the process has parametric uncertainties, it is intuitive that the choice of P can be made with at least a real pole dominant with respect to each couple of complex poles and then to go on by using the Theorems of Sturm and/or Kharitonov or with new results or directly with the command roots and with the Monte Carlo method.Regarding this the following main theorem holds.Theorem 8. Give the process (3) with limited uncertainties and with assigned nominal values of its parameters.Suppose that there exists a set of reference poles  22), ( 24), ( 25) and ( 26), for ρ big enough it is ˆ() () () () ds ds ds h ns ≅=+ # . From this the proof easily follows.In the following, for brevity, the second, third, fourth-order control systems will be considered.
Theorem 10.Let be  then the poles of () Ws are all real or the real pole is on the right of the remaining couple of complex poles, i.e. the system is externally positive.Proof.Let be 123 ,, p pp the poles of () Ws note that the "barycentre"   with a real negative root on the right of the remaining complex roots (see Figs. 9, 10), the proof easily follows. where under the hypothesis that the initial state of the control system of Fig. 2, with 13 c nn =+= and îi kk = , is null and that (0) (0) 0 rd − = , the error () et of the control system of Fig. 2, considering all the possible values of the process, satisfies relation Under the hypothesis that each activation system is an electric DC motor (with inertial load, possible resistance in series and negligble inductance of armature) powered by using a power amplifier, the model of the process turns out to be By choosing max ( , , ) 1.436e5 0, max ( , , ) 1.454e5 0  12 ).In Fig. 11 the time histories of x r , x r $ and, under the hypothesis that ˆ2 v K = , the corresponding error x e , in accordance with the proposed results are reported.Clearly the "tracking precision" is unchanged   Part II

Problem formulation and preliminary results
Now consider the following class of nonlinear dynamic system , where tR ∈ is the time, is the uncertain parametric vector of the system, with ℘ a compact set, 1 mr FR × ∈ is a limited matrix with rank m , 2 mxm i FR ∈ are limited matrices and m f R ∈ is a limited vector which models possible disturbances and/or particular nonlinearities of the system.In the following it is supposed that there exists at least a matrix ( , ) with ℘ a compact set, is the vector of the uncertain parameters of the mechanical system, -(,, ), cC q q p q = $$ with C linear with respect to q $ , is the vector of the generalized centrifugal forces, the Coriolis and friction ones, -(, , ) gg t q p = is the vector of the generalized gravitational and elastic forces and of the external disturbances, u is the vector of the generalized control forces produced by the actuators, -T is the transmission matrix of the generalized control forces.If system (56) is controlled by using the following state feedback control law with a partial compensation ( )   Proof.The proof of (61) easily follows by solving, with the use of the method of separation of variables, the equation Proof.The proof can be found in (Celentano, 2010).

Main results
Now the following main result, which provides a majorant system of the considered control system, is stated.( ) By choosing a matrix 0 0 From ( 69), ( 81) and (82) the relation (74) easily follows.Relations (75) easily follow from the third of (59), from (70), from the third of (72) and by considering (78).Remark 10.It is easy to note that the values of c and p c provided by ( 75) are the same if, instead of y and y $ , their components i y e i y $ are considered.Now the main result can be stated.It allows determining the control law which guarantees prefixed majorant values of the time constant   Remark 12.The stated theorems can be used for determining simple and robust control laws of the PD type, with a possible compensation action, in order to force system (56) to track a www.intechopen.com New Robust Tracking and Stabilization Methods for Significant Classes of Uncertain Linear and Nonlinear Systems 269 generic preassigned limited in "acceleration" trajectory, with preassigned increases of the maximum "position and/or velocity" errors and preassigned increases of the time constants characterizing the convergence of the error.

Conclusion
In this chapter it is has been considered one of the main and most realistic control problem not suitable solved in literature (to design robust control laws to force an uncertain parametric system subject to disturbances to track generic references but regular enough with a maximum prefixed error starting from a prefixed instant time).This problem is satisfactorily solved for SISO processes, without zeros, with measurable state and with parametric uncertainties by using theorems and algorithms deriving from some proprierties of the most common filters, from Kharitonov's theorem and from the theory of the externally positive systems.
The considered problem has been solved also for a class of uncertain pseudo-quadratic systems, including articulated mechanical ones, but for limitation of pages only the two fundamental results have been reported.They allow to calculate, by using efficient algorithms, the parameters characterizing the performances of the control system as a function of the design parameters of the control law.

Fig. 2 .
Fig. 2. State feedback control scheme with an I control action.Remark 3. It is useful to note that often the state-space model of the process (1) is already in the corresponding companion form of the input-output model of the system (3) (think to the case in which this model is obtained experimentally by using e.g.Matlab command invfreqs); on the contrary, it is easy to transform the interval uncertainties of ,, ABC into the ones (even if more conservative) of , i ab .
Theorem 9. S o m e s e t s o f r e f e r e n c e p o l e s P which satisfy Theorem 8 are: ω such that the roots of () ns are real (e.g. Fig. 5. Root locus of () ,

Fig. 7 .
Fig. 7. δ in the case of real pole on the right of the couple of complex poles.

Fig. 8 .
Fig. 8. δ in the case of all real poles.

Fig. 9 .
Fig. 9. Root locus of the polynomial (46) in the hypothesis that all the roots of ( ) ds − are real.

Fig. 10 .
Fig. 10.Root locus of the polynomial (46) under the hypothesis that ( ) ds − has a real negative root on the right of the remaining complex roots.Finally, from Theorems 7, 9, 11 and by using the Routh criterion the next theorem easily follows.Theorem 12. Give the process (3) with limited uncertainties for 13 c nn = += and assigned some design values of ˆv K and ˆrd δ − $ $.Let be choose

8 .
ds , it is possible to satisfy also the specification about max τ ; so the specifications 2. are all satisfied.Remark Give the process (3) with limited uncertainties and assigne the design values of ˆv K , by choosing P in accordance with Theorem 9, a controller such that, for all the possible values of the parameters of the process, max max τ τ ≤ and the error () et satisfies relation (2), can be obtained by increasing, if necessary, iteratively ρ starting from the value of the command roots and with the Monte Carlo method.According to this, note that for 4 c n ≤ the control system of Fig.2(for an assigned set of parameters) is externally positive and proposed design method, by taking into account Theorem 8, can be easily extended in the case of 4 c n ≥ .Example 1.Consider a planar robot (e.g. a plotter) whose end-effector must plot dashed linear and continuous lines with constant velocities during each line.

Fig. 11 .
Fig. 11.Time histories of , xx rr $ and x e for ˆ2 v K = .
τ is the time constant of the linearized of system (60) and 20 ρ is the upper bound of the convergence interval of () t ρ for 0 d = , i.e. of system (60) in free evolution.

γ
Fig. 16.Time history of ρ and γ as a function of 0 ρ

α
As regards the determination of K in order to satisfy the first of (84), the computation of c u to decrease d and regarding the computation of 2 α and d , for limitation of pages, it has to be noted at least that for the mechanical systems, that the inertia matrix B is symmetric and p.d. and d can be facilitated by suitably using Theorem 15.
By using again the change of scale property of the Laplace transform, by taking into account (10) and (11) it is www.intechopen.com robustness of the constant gain and of the maximum time constant with respect to the parametric uncertainties of the process).
, , 1, 2 ,..., , www.intechopen.com Note that, if the uncertainties of the process are small enough and ρ is chosen big It is important to note that the class of systems (56) includes the one, very important from a practical point of view, of the articulated mechanical systems (mechanical structures, flexible too, robots,…).Indeed it is well-known that mechanical systems can be described as follows www.intechopen.com ∀> , it follows that P is p.d. and, therefore, also P is p.d. .