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Control theory for linear systems

control systems theory and applications for linear repetitive processes geometric control theory for linear systems a tutorial
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Control theory for linear systems Harry L. Trentelman Research Institute of Mathematics and Computer Science University of Groningen P.O. Box 800, 9700 AV Groningen The Netherlands Tel. +31-50-3633998 Fax. +31-50-3633976 E-mail. h.l.trentelmanmath.rug.nl Anton A. Stoorvogel Dept. of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands Tel. +31-40-2472378 Fax. +31-40-2442489 E-mail. A.A.Stoorvogeltue.nl Malo Hautus Dept. of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands Tel. +31-40-2472628 Fax. +31-40-2442489 E-mail. M.L.J.Hautustue.nl May 15, 2002Chapter 1 Introduction 1.1 Control system design and mathematical control theory Very roughly speaking, control system design deals with the problem of making a concrete physical system behave according to certain desired specifications. The ul- timate product of a control system design problem is a physical device that, if con- nected to the to be controlled physical system, makes it behave according to the spec- ifications. This device is called a controller. To get from a concrete to be controlled physical system to a concrete physical device to control the system, the following intermediate steps are often taken. First, a mathematical model of the physical system is made. Such a mathematical model can take many forms. For example, the model could be in the form of a system of ordinary and/or partial differential equations, together with a number of algebraic equations, relating the relevant variables of the system. The model could also involve difference equations, some of the variables could be related by transfer functions, etc. The usual way to get a model of an actual system is to apply the basic laws that the system satisfies. Often, this method is called first principles modeling. For example, if one deals with an electro-mechanical system, the set of basic physical laws (Newton’s laws, Kirchoff’s laws, etc.) that variables in the system satisfy form a mathematical model. A second way to get a model is called system identification. Here, the idea is to do experiments on the physical system: certain variables in the physical system are set to particular values from the outside, and at the same time other variables are measured. In this way, one tries to estimate (’identify’) the laws that the variables in the system satisfy, thus obtaining a model. Very often, a combination of first principles modeling and system identification is used to obtain a model. The second step in a control system design problem is to decide which desirable properties we want the physical system to satisfy. Very often, these properties can be formulated mathematically by requiring the mathematical model to have certain2 Introduction qualitative or quantitative mathematical properties. Together, these properties form the design specifications. The third, very crucial, step is to design, on the basis of the mathematical model of the physical system, and the list of design specifications, a mathematical model of the physical controller device. It is this step in the control design problem that we deal with in this book: it deals with mathematical control theory, in other words, with the mathematical theory of design of models of controllers. The problem of getting from a model, and a list of design specifications to a model of a controller is called a control synthesis problem. Of course, for a given model, each particular list of design specifications will give rise to a particular control synthesis problem. In this book we will study for a great variety of design specifications the corresponding control synthesis problems. We restrict ourselves in this book to a particular class of mathematical models: we assume that our models (both of the physical, to be controlled systems, as well as the controllers) are linear, time-invariant, finite-dimensional state-space systems with inputs and outputs. This class of models is rich enough to treat the fundamental issues in control system design, and the resulting design techniques work remarkably well for a large class of concrete control system design problems encountered in engineering applications. A final step in the control system design problem is, of course, to realize the mathematical model of the controller by an actual physical device, often in the form of suitable hardware and software, and to interconnect this device with the to be controlled physical system. As an illustration of a control design problem for a concrete physical system, we consider the motion of a communications satellite. In order for a satellite to have a fixed position to an observer on the earth’s surface, while moving with its jet engines switched off, it has to describe a circular orbit, say in the equator plane, at a fixed altitude of 35620 km, with the same velocity of rotation as the earth (this orbit of a satellite is called a geostationary orbit). We wish to be able to influence the motion of the satellite such that it remains in this geostationary orbit. In order to do this, we want to build a device that exerts forces on the satellite when needed, by means of the satellite’s jet-engines. In the actual control design problem, this physical system should first be described by a mathematical model. In this example, a first mathematical model of the satellite’s motion (based on the assumption that the satellite is represented by a point mass) will consist of a set of non-linear differential equations that can be deduced using elementary laws from physics. From this model, we can obtain a simplified one in the form of a linear, time-invariant, finite-dimensional state-space system with inputs and outputs. In this simplified model, the geostationary orbit corresponds to the zero equilib- rium solution of this finite-dimensional state-space system. Of course, if initially the satellite is placed in this equilibrium position, then it will remain there for ever, as desired. However, if, for some reason, at some moment in time the position of the satellite is perturbed slightly, then it will from that moment on follow a trajectoryControl system design and mathematical control theory 3 corresponding to an undesired periodic motion in the equator plane, away from the equilibrium solution. Our desire is to design a controller such that the equilibrium so- lution corresponding to the geostationary orbit becomes asymptotically stable. This would guarantee that trajectories starting in small perturbations away from the equi- librium solution converge back to that equilibrium solution as time runs off to infinity. In other words, the design specification is: asymptotic stability of the equilibrium so- lution of the controlled system. Based on the linear, time invariant, finite-dimensional state-space model of the satellite’s motion around the geostationary orbit and on the design specification of asymptotic stability, the next step is to find a model of a controller that achieves the design specification. The controller should also be chosen from the class of linear, time-invariant, finite-dimensional state-space systems with inputs and outputs. In mathematical control theory, the mathematical model describing the physical system that we want to behave according to the specifications is called the control system, the system to be controlled,or the plant, and the mathematical model of the controller device that is aimed at achieving these specifications is called the controller. The mathematical description of the system to be controlled, together with the controller is called the controlled system. In our example, we want the controlled system to be asymptotically stable. An important paradigm in control systems design, and in mathematical control theory, is feedback. The idea of feedback is to let the action of the physical con- trolling device at any moment in time depend on the actual behavior of the physical system that is being controlled. This idea imposes a certain ‘smart’ structure on the controlling device: it ‘looks’ at the system that it is influencing, and decides on the basis of what it ‘sees’ how it will influence the system the next moment. In the ex- ample of the communications satellite, the controlling device, at a fixed position on the earth’s surface, takes a measurement of the position of the satellite. Depending on the deviation from the desired fixed position, the controlling device then exerts certain forces on the satellite by switching on or off its jet-engines using radio signals. Any physical controller device that has this feedback structure is called a feedback controller. In terms of its mathematical model, the feedback structure of a controller is often represented by certain variables (representing what the controller ‘sees’) be- ing mapped to other variables (representing the influence that the controller exerts on the system). The first kind of variables are called measured outputs of the system, the second kind of variables are called control inputs to the system. Typically, the input variables are considered to be caused by the measured outputs. Mathematically, the relation between the measured outputs and the control inputs can be described by a map. Often, designing a controller for a given system can be formulated as the pro- blem of finding a suitable map between measured outputs and control inputs. The controlled system corresponding to the combination of a control system and a feed- back controller is often called the closed-loop system, and one often speaks about the interconnection of the control system and the feedback controller. The principle of feedback is illustrated pictorially in the diagram in Figure 1.1 on the next page. The control synthesis problems treated in this book are all concerned with the4 Introduction  . . System to . . . .  be controlled  control measured inputs outputs   . Feedback . . . . . controller  Figure 1.1: The principle of feedback design of feedback controllers: given a linear time-invariant state-space system, and certain design specifications, find a feedback controller such that the design spec- ifications are fulfilled by the closed-loop system, or determine that such feedback controller does not exist. 1.2 An example: instability of the geostationary orbit In this example we will take a more detailed look at the system describing the motion of a communications satellite. The principle of such a satellite is that it serves as a mirror for electromagnetic signals. In order not to be forced to continuously aim the transmitters and receiving antennas at the satellite, it is desired that the satellite has a fixed position with respect to these devices. This also has the advantage that the satellite does not go down and rise, so that it can be used 24 hours a day. In order to simplify the example, we will consider the motion of the satellite in the equator plane. By taking the origin at the center of the earth, the position of the satellite is given by its polar coordinates (r,θ). Introduce the following constants: M := mass of the earth, E G:= earth’s gravitational constant, := earth’s angular velocity, M := mass of the satellite. S We assume that the satellite has on-board jets which make it possible to exert forces F (t) and F (t) to the satellite in the direction of r and θ, respectively. Using New- r θ ton’s law it can be verified that the equations of motion of the satellite are given by GM F (t) 2 E r ˙ r¨(t)= r(t)θ(t) − + , 2 M r(t) S ˙ r˙(t)θ(t) F (t) θ ¨ θ(t)=−2 + . r(t) M r(t) SAn example: instability of the geostationary orbit 5 The desired geostationary orbit is given by θ(t)= θ + t, 0 r(t)= R , 0 F (t)= 0, r F (t)= 0, θ where R still has to be determined. We first check that this indeed yields a solution to 0 the equations of motion for suitable R . By substitution in the differential equations 0 we obtain GM E 2 0= R  − , 0 2 R 0 which yields  GM 3 E R = . 0 2  By taking the appropriate values for the physical constants in this formula, we find that R is approximately equal to 42000 km. Thus the geostationary orbit is a circular 0 orbit in the equator plane, at an altitude of approximately 35620 km over the equator (the radius of the earth being approximately 6380 km). It is convenient to replace the equations of motions by an equivalent system of four first order differential equations by putting x := r(t)− R , 1 0 x := r˙(t), 2 x := θ(t)− (θ + t), 3 0 ˙ x := θ− . 4 Note that x is the deviation from the desired angle. This value can at any time 3 instant be measured by an observer on the equator by comparing the actual position of the satellite to the desired position. In terms of these new variables, the system is described by     x (t) 2 x˙ (t) 1 GM F (t)  2 E r    (x (t)+ R )(x (t)+ ) − + x˙ (t) 1 0 4  2  2 M (x (t)+R ) S   1 0 = (1.1)     x˙ (t) x (t) 3   4 2x (t)(x (t)+) F (t) θ 2 4 x˙ (t) 4 − + x (t)+R M (x (t)+R ) S 1 0 1 0 The equations (1.1) constitute a nonlinear state-space model with inputs and outputs. T The control input is u= (F , F ) , for the measured output one could take x . The r θ 3 geostationary orbit corresponds to the equilibrium solution given by (x , x , x , x )= (0, 0, 0, 0), F = 0, F = 0. 1 2 3 4 r θ6 Introduction By Kepler’s law it is clear that if at time t the equilibrium solution is perturbed to, 0 say, (x (t ), x (t ), x (t ), x (t ))= (ξ ,ξ ,ξ ,ξ ), 1 0 2 0 3 0 4 0 1 2 3 4 then the resulting orbit will be an ellipsoid in the equator plane with the earth in one of its focuses. The angular velocity of the satellite with respect to the center of earth will then no longer be constant, so to an observer on the equator the satellite will not be in a fixed position, but will actually go down and rise periodically. Mathe- matically this can be expressed by saying that the equilibrium solution is not locally asymptotically stable. What can we do about this? We still have the possibility to exert forces to the satellite in the r-direction and in the θ-direction. Also, the variable x can be measured. The control synthesis problem can now be formulated as: find 3 T a feedback controller that generates a control input u= (F , F ) on the basis of the r θ measured output x , in such a way that the equilibrium solution corresponding to the 3 geostationary orbit becomes locally asymptotically stable. Of course, it is not clear a priori whether such controller exists. system  F r  x 3 F θ  ? 1.3 Linear control systems The above is an example in which the system to be controlled is a nonlinear system. T The vector (x , x , x , x ) is called the state variable of the system. Solutions to the 1 2 3 4 4 differential equations (1.1) take their values in the state spaceR . The control input T 2 u= (F , F ) takes its values in the input spaceR , the measured output x takes its r θ 3 values in the output spaceR. More generally, a control system with state variable x, n m state spaceR , input variable u, input spaceR , output variable y, and output space p R is given by the following equations: x˙(t)= f (x(t), u(t)), y(t)= g(x(t), u(t)). n m n n m p Here, f is a function fromR ×R toR and g is a function fromR ×R toR . If f and g are linear, then we obtain a linear control system.If f is linear then there n n m n exist linear maps A:R →R and B:R →R such that f (x, u)= Ax+ Bu.IfExample: linearization around the geostationary orbit 7 n p m p g is linear then there exist linear maps C : R → R and D : R → R such that g(x, u)= Cx+ Du. The equations of the corresponding control system are then x˙(t)= Ax(t)+ Bu(t), y(t)= Cx(t)+ Du(t). This is called a linear, time-invariant, finite-dimensional state-space system. In this book we will exclusively deal with the latter kind of control system models. Many real life systems can be modeled very well by this kind of system models. Often, the behavior of a nonlinear system can, at least in the neighborhood of an equilibrium solution, be approximately modeled by such a linear system. 1.4 Example: linearization around the geostationary orbit Again consider the motion of the satellite. We arrived at a nonlinear control system described by the equations x˙(t)= f (x(t), u(t)), (1.2) y(t)= g(x(t)), T T where u = (F , F ) is the control input, x = (x , x , x , x ) is the state variable, r θ 1 2 3 4 4 2 4 and where y denotes the measured output x . The function f : R ×R → R is 3 given by   x 2 GM F  2 E r  (x + R )(x + ) − + 1 0 4 2   T T M (x +R ) S 1 0 f ((x , x , x , x ) ,(F , F ) )= ,   1 2 3 4 r θ x   4 2x (x +) F 2 4 θ − + x +R M (x +R ) 1 0 S 1 0 4 and the function g : R → R is simply given by g(x , x , x , x ) = x . Using a 1 2 3 4 3 Taylor expansion in a neighborhood of 0, we know that for x and F small f (x, u)≈ f (0, 0)+ D f (0, 0)x+ D f (0, 0)u. x u T Here, D f and D f are the derivatives of f with respect to x= (x , x , x , x ) and x u 1 2 3 4 T u= (F , F ) . In our example we have f (0, 0)= 0 and r θ   01 0 0 2   3 002R 0   D f (0, 0)= , x   00 0 1 2 0 − 00 R 0   00 1   0 M  S  D f (0, 0)= . u   00 1 0 M R S 08 Introduction GM E 2 T Here, we have used the fact that =  . Thus, for small x = (x , x , x , x ) 1 2 3 4 3 R 0 T and u= (F , F ) , the original nonlinear control system can be approximated by the r θ linear control system x˙ (t)= x (t), 1 2 F 2 r x˙ (t)= 3 x (t)+ 2R x (t)+ , 2 1 0 4 M S (1.3) x˙ (t)= x (t), 3 4 2 F θ x˙ (t)=− x (t)+ . 4 2 R M R 0 S 0 Of course this can be written as x˙(t)= Ax(t)+ Bu(t), (1.4) y(t)= Cx(t), with A:= D f (0, 0), B:= D f (0, 0), and C:= (00 10). This linear control sys- x u tem is called the linearization of the original system around the equilibrium solution (u, x, y)= (0, 0, 0). 1.5 Linear controllers As explained, a feedback controller for a given control system is a mathematical model that generates control input signals for the system to be controlled on the basis of measured outputs of this system. If we are dealing with a system in state space form given by the equations x˙(t)= f (x(t), u(t)), (1.5) y(t)= g(x(t), u(t)), then a possible choice for the form of such a mathematical model is to mimic the form of the control system, and to consider pairs of equations of the form w( ˙ t)= h(w(t), y(t)), (1.6) u(t)= k(w(t), y(t)). Any such pair of equations will be called a feedback controller for the system (1.5).  The variable w is called the state variable of the controller, it takes its values inR for some . The controller is completely determined by the integer , together with the functions h and k. The measured output y is taken as an input for the controller. On the basis of y the controller determines the control input u. If we are dealing with a linear control system given by the equations x˙(t)= Ax(t)+ Bu(t), y(t)= Cx(t)+ Du(t), then it is reasonable to consider feedback controllers of a form that is compatible with the linearity of these equations. This means that we will consider controllers ofExample: regulation of the satellite’s position 9 the form (1.6) in which the functions h and k are linear. Such controllers are also represented by linear, time-invariant, finite-dimensional systems in state space form, given by w( ˙ t)= Kw(t)+ Ly(t), (1.7) u(t)= Mw(t)+ Ny(t), where K, L, M and N are linear maps. The state variable of the controller is w.Any pair of equations (1.7) is called a linear feedback controller . 1.6 Example: stabilizing the geostationary orbit Our design specification is local asymptotic stability of the geostationary orbit. With- out going into the details, we mention the following important result on local asymp- totic stability of a given stationary solution of a system of first order nonlinear differ- ential equations: if the linearization around the stationary solution is asymptotically stable, then the stationary solution itself is locally asymptotically stable. This means that if we succeed in finding a linear controller for the linearization (1.4), then the same linear controller applied to the original nonlinear control system will make the geostationary orbit locally asymptotically stable Consider the linearization (1.4) around this stationary solution. If we interconnect this linear system with a linear controller of the form, w( ˙ t)= Kw(t)+ Ly(t), (1.8) u(t)= Mw(t)+ Ny(t), then the resulting closed-loop system is obtained by substituting u= Mw+ Ny into (1.4) and y= Cx into (1.8). This yields    x˙(t) A+ BNC BM x(t) = . (1.9) w( ˙ t) LC K w(t) This system of first order linear differential equations is asymptotically stable if and only if all eigenvalues λ of the matrix  A+ BNC BM A := e LC K satisfye λ 0, i.e., have negative real parts. A matrix with this property is referred to as a stability matrix. The problem is to find matrices K, L, M and N such that this property holds. Once we have found these matrices, the corresponding linear controller applied to the original nonlinear system will make the geostationary orbit locally asymptotically stable. Indeed, the interconnection of (1.2) with (1.8) is given by x˙(t)= f (x(t), Mw(t)+ Ng(x(t))), w( ˙ t)= Kw(t)+ Lg(x(t)), and its linearization around the stationary solution (0, 0) is exactly given by (1.9).10 Introduction 1.7 Example: regulation of the satellite’s position In the previous section we discussed the problem of making the geostationary orbit T asymptotically stable. This property guarantees that the state x = (x , x , x , x ) , 1 2 3 4 T after an initial perturbation x(0)= (ξ ,ξ ,ξ ,ξ ) away from the zero equilibrium, 1 2 3 4 converges back to zero as time runs off to infinity. In the satellite example, we are very much interested in two particular variables, namely x (t)= r(t)− R and x (t)= 1 0 3 θ(t)−(θ +t). The values of these variables express the deviation from the satellite’s 0 required position. It is very important that, after a possible initial perturbation away from the zero solution, these values return to zero as quickly as possible. The design specification of asymptotic stability alone is too weak to achieve this quick return to zero. This motivates the following approach. Given the linear model (1.3) of the satel- lite’s motion around the geostationary equilibrium and a possible perturbation x(0)= T ξ = (ξ ,ξ ,ξ ,ξ ) away from the equilibrium, express the performance of the sys- 1 2 3 4 T tem by the following functional of the control input u= (F , F ) : r θ ∞ 2 2 2 2 J(ξ, F , F )= αx (t)+ βx (t)+ γ F (t)+ δF (t)dt (1.10) r θ 1 2 r θ 0 Here, α, β, γ , and δ are non-negative constants, called weighting coefficients.A reasonable control synthesis problem is now to design a feedback controller that gen- T erates input functions u = (F , F ) , for example on the basis of measurements r θ T of the state variables x = (x , x , x , x ) , such that the performance functional 1 2 2 4 J(ξ, F , F ) is as small as possible, while at the same time the closed-loop system is r θ asymptotically stable. More concretely, one could try to find a feedback controller of the form u= Fx, 4 2 with F a linear map from R to R that has to be determined, such that (1.10) is minimal, and such that the closed-loop system is asymptotically stable. Such feed- back controller where u is a static linear function of the state variable x is called a static state feedback control law. By a suitable choice of the weighting coefficients, ∞ ∞ 2 2 it is expected that in the closed-loop system both x (t)dt and x (t)dt are 0 1 0 3 small, so that x (t) and x (t) will return to a small neighborhood of zero quickly, as 1 3 desired. A feedback controller that minimizes the quadratic performance functional (1.10) is called a linear quadratic regulator. This terminology comes from the fact that the underlying control system is linear, and the performance functional depends quadratically on the state and input variables. 1.8 Exogenous inputs and outputs to be controlled Often, if we make a mathematical model of a real life physical system, we do not only want to specify the control inputs, but also a second kind of inputs, the exogenousExample: including the moon’s gravitational field 11 inputs. These can be used, for example, to model unknown disturbances that act on the system. Also, they can be used to ‘inject’ into the system the description of given time functions that certain variables in the system are required to track. In this case the exogenous inputs are called reference signals. Often, apart from the measured output, we want to include in our mathematical model a second kind of outputs, the outputs to be controlled, also called the exogenous outputs. Typically, the outputs to be controlled include those variables in the control system that we are particularly interested in, and that, for example, we want to keep close to certain, a priori given, values. A general control system in state space form with exogenous inputs and outputs to be controlled is described by the following equations: x˙(t)= f (x(t), u(t), d(t)), y(t)= g(x(t), u(t), d(t)), (1.11) z(t)= h(x(t), u(t), d(t)). Here, d represents the exogenous inputs. The functions d are assumed to take their r values in some fixed finite-dimensional linear space, say,R . The variable z repre- sents the outputs to be controlled, which are assumed to take their values in, say, q R . The variables x, u and y are as before, and the functions f, g and h are smooth functions mapping between the appropriately dimensioned linear spaces. Typically, the function h is chosen in such a way that z represents those variables in the system ∗ that we want to keep close to, or at, some prespecified value z , regardless of the disturbance inputs d that happen to act on the system. Again, if f , g and h are linear functions, then the equations take the following form x˙(t)= Ax(t)+ Bu(t)+ Ed(t), z(t)= C x(t)+ D u(t)+ D d(t), 1 11 12 y(t)= C x(t)+ D u(t)+ D d(t), 2 21 22 for given linear maps A, B, E, C , D , D , C , D and D . These equations are 1 11 12 2 21 22 said to constitute a linear control system in state space form with exogenous inputs and outputs. Many real life systems can be modelled quite satisfactorily in this way. Moreover, the behavior of nonlinear systems around equilibrium solutions is often modelled by such linear systems. 1.9 Example: including the moon’s gravitational field In the equations of motion of our satellite we did not include gravitational forces acting on the satellite caused by other bodies than the earth. Now suppose that in our satellite model we want to include the forces caused by the gravitational field of the moon. We can do this by including into our system the forces exerted by the moon on the satellite as disturbance inputs, whose values are unknown. Let F and F be M,r M,θ the forces applied by the moon in the r and θ direction, respectively. Including these12 Introduction into the model, we obtain     x (t) 2 x˙ (t) 1 F (t) GM F (t) M,r  2 E r    (x (t)+ R )(x (t)+ ) − + + x˙ (t) 1 0 4  2  2 M M (x (t)+R ) S S   1 0 = (1.12)     x˙ (t) x (t) 3   4 F (t) 2x (t)(x (t)+) F (t) M,θ 2 4 θ x˙ (t) 4 − + + x (t)+R M (x (t)+R ) M (x (t)+R ) 1 0 S 1 0 S 1 0 We are particularly interested in the variables x (t) and x (t), describing the deviation 1 3 from the desired geostationary orbit (R ,θ +t). Thus, as output to be controlled we 0 0 can take the vector (x , x ). In this way we exactly obtain a model of the form (1.11), 1 3 T T with control input u = (F , F ) , exogenous input d = (F , F ) , measured r θ M,r M,θ T output y= x , and output to be controlled z= (x , x ) . 3 1 3 Of course, an equilibrium solution is given by (x , x , x , x )= (0, 0, 0, 0), 1 2 3 4 (F , F )= (0, 0), r θ (F , F )= (0, 0). M,r M,θ By linearization around this stationary solution, we find that for small (x , x , x , x ), 1 2 3 4 small (F , F ) and small (F , F ), our original control system is approximated r θ M,r M,θ by x˙ (t)= x (t), 1 2 F (t) F (t) 2 r M,r x˙ (t)= 3 x (t)+ 2R x (t)+ + , 2 1 0 4 M M S S x˙ (t)= x (t), 3 4 F (t) 2 F (t) M,θ θ x˙ (t)=− x (t)+ + .y(t)= x (t), 4 2 3 R M R M R  0 S 0 S 0 x (t) 1 z(t)= . x (t) 3 Of course, these equations constitute a linear control system in state space form with exogenous inputs and outputs: x˙(t)= Ax(t)+ Bu(t)+ Ed(t), y(t)= C x(t), (1.13) 1 z(t)= C x(t), 2 with A and B as before, E:= B, C := 00 10 and 1  10 00 C := . 2 00 10 If we interconnect this system with a linear controller of the form (1.8), then the closed-loop system will be given by     x˙(t) A+ BNC BM x(t) E 1 = + d(t), w( ˙ t) LC K w(t) 0 1  (1.14) x(t) z(t) = C 0 . 2 w(t)Notes and references 13 Our control synthesis problem might now be to invent a linear controller such that, in the closed-loop system, the disturbance input d (= (F , F )) does not influ- M,r M,θ ence the output z. If a controller achieves this design specification, then it is said to achieve disturbance decoupling. If this design specification is fulfilled, then, at least according to the linear model, the satellite will remain in its geostationary or- bit once it has been put there, regardless of the gravitational forces of the moon. Of course, part of the design problem would be to answer the question whether such con- troller actually exists. If it does not exist, one could weaken the design specification, and require that the influence of the disturbances on the outputs to be controlled be as small as possible, in some appropriate sense. One could, of course, also ask for combinations of design specifications to be satisfied, for example both disturbance decoupling and asymptotic stability of the closed-loop system, in the sense of section 1.6. Alternatively, one could try to design a controller that makes the influence of the disturbances on the output to be controlled as small as possible, while making the closed-loop system asymptotically stable, again in the sense of section 1.6. 1.10 Robust stabilization In general, a mathematical model of a real life physical system is based on many idealizing assumptions. Thus, in general, the control system that models a certain real life phenomenon will not be a precise description of that phenomenon. Thus it might happen that a controller that asymptotically stabilizes the control system that we are working with, does not make the real life system behave in a stable way at all, simply because the control system we are working with is not a good description of this real life system. Sometimes, it is not unreasonable to assume that the correct description rather lies in a neighborhood (in some appropriate sense) of the control system that we are working with (this control system is often called the nominal system). In order to assure that a controller also stabilizes our real life system, we could formulate the following design specification: given the nominal control system, together with a fixed neighborhood of this system, find a controller that stabilizes all systems in that neighborhood. If a controller achieves this design objective, we say that it robustly stabilizes the nominal system. As an example, consider the linear control system that models the motion of the satellite around its stationary solution. This model was obtained under several ideal- izing assumptions. For example, we have neglected the dynamics of the satellite that are caused by the fact that, in reality, it is not a point mass. If these additional dynam- ics were taken into account in the nonlinear control system, then we would obtain a different linearization, lying in a neighborhood (in an appropriate sense) of the orig- inal (nominal) linearization, described in section 1.4. One could then try to design a robustly stabilizing controller for the nominal linearization. Such controller will not only stabilize the nominal control system, but also all systems in a neighborhood of the nominal one.14 Introduction 1.11 Notes and references Many textbooks on control systems design, and the mathematical theory of systems and control are available. Among the more recent engineering oriented textbooks we mention the books by Van de Vegte 202, Phillips and Harbor 145, Franklin, Power and Emami-Naeini 49, and Kuo 101. Among the textbooks that concentrate more on the mathematical aspects of systems and control we mention the classical textbooks by Kwakernaak and Sivan 105 and Brockett 25. The satellite example that was discussed in this chapter is a standard example in several textbooks; see for instance the book by Brockett 25. We also mention the seminal book by Wonham 223, which was the main source of inspiration for the geometric ideas and methods used in this book. Other relevant textbooks are the books by Kailath 90, Sontag 181, Maciejowski 118, and Doyle, Francis and Tannenbaum, 40. As more more recent textbooks on control theory for linear systems we mention the books by Green and Limebeer 66, Zhou, Doyle and Glover 232, and Dullerud and Paganini 42. For textbooks on system identification and modelling we would like to refer to the books by Ljung 112 and Ljung and Glad 113. For textbooks on systems and control theory for nonlinear systems, we refer to the books by Isidori 86, Nijmeijer and Van der Schaft 134, Khalil 99, and Vidyasagar 208.Chapter 2 Mathematical preliminaries In this chapter we start from the assumption that the reader is familiar with the concept of vector spaces (or linear spaces) and with linear maps. The objective of this chapter is to give a short summary of the standard linear-algebra tools to be used in this book with special emphasis on the geometric (as opposed to matrix) properties. 2.1 Linear spaces and subspaces Linear spaces are typically denoted by script symbols likeV,X,.... We will only be dealing with finite-dimensional spaces. LetX be a linear space andV,W subspaces ofX. ThenV∩W andV+W :=x+ y x ∈ V, y ∈ W are also subspaces. The diagram in Figure 2.1 symbolizes the various inclusion relations between these spaces. It is easily seen thatV+W is the smallest subspace containing bothV and X V+W V W V∩W 0 Figure 2.1: Lattice diagram16 Mathematical preliminaries W, i.e., if a subspaceL satisfiesV⊂L andW ⊂L thenV+W ⊂L. Similarly, V∩W is the largest subspace contained in bothV andW. The fact that for every pair of subspaces there exists a smallest subspace containing both subspaces and a largest subspace contained in both spaces, is expressed by saying that the set of subspaces of X forms a lattice, and diagram in Figure 2.1 is called a lattice diagram. IfV,W andR are subspaces andV⊂R then R∩ (V+W)=V+ (R∩W). (2.1) This formula, which can be proved by direct verification (see exercise 2.1), is called the modular rule. LetV ,V ,...,V be subspaces. Then they are called (linearly) 1 2 k independent if every x ∈ V +V +···+V has a unique representation of the 1 2 k form x = x + x +···+ x with x ∈ V (i = 1,..., k), equivalently, if x ∈ 1 2 k i i i V (i = 1,..., k) and x +···+ x = 0 imply x =···= x = 0. Still another i 1 k 1 k characterization is  V ∩ V = 0 (i= 1,..., k). (2.2) i j j =i Here, the symbol 0 is used to denote the null subspace of a vector space, i.e. the subspace consisting only of the element 0. IfV ,...,V are independent subspaces, 1 k their sumV is called the direct sum ofV ,...,V and it is written 1 k k  V=V ⊕···⊕V = V . 1 k k i=1 IfV is a subspace then there exists a subspaceW such thatV⊕W = X. Such a subspace is called a (linear) complement ofV. It can be constructed by first choosing a basis q ,..., q ofV and then extending it to a basis q ,..., q ofX. Then the 1 k 1 n span of q ,..., q is a complement ofV, as can easily be verified. Obviously, a k+1 n complement is not unique. The linear spaces we are interested in are spaces over the field R of real num- bers. For some purposes, however, it is convenient to allow also complex vectors and coefficients. We denote by C the field of complex numbers. We use the complex extensionX of a given linear spaceX, consisting of all vectors of the form v+ iw, C where v and w are inX. Many statements made in terms ofX can easily be trans- C lated to corresponding results aboutX. In this book, we will freely use the complex extension, often without explicitly mentioning it. 2.2 Linear maps For a linear map A:X→Y, we define ker A:=x∈X Ax= 0, (2.3) im A :=Ax x∈X,Linear maps 17 called the kernel and image of A, respectively. These are linear spaces. We say that A is surjective if im A=Y and injective if ker A= 0. Also, A is called bijective (or an isomorphism)if A is injective and surjective. In this case, A has an inverse map, −1 usually denoted by A . It is also known that a bijection A:X→Y exists if and only if dimX= dimY (supposing, as we always do, that the linear spaces are finite dimensional). In general, if A:X→Y is a, not necessarily invertible, linear map and ifV is a subspace ofY, then the inverse image ofV is the subspace ofX defined by −1 A V:=x∈X Ax∈V Given A : X → X and a subspaceV ofX, we say thatV is A-invariant (or, if the map A is supposed to be obvious, simply invariant) if for all x ∈V we have Ax ∈ V, which can be written as AV ⊂ V. The concept of invariance will play a crucial role in this book. One can consider various types of restriction of a linear map: • If B : U→ X is a linear map satisfying im B ⊂ V whereV is a subspace ofX then the codomain restriction of B is the map B : U → V satisfying Bu:= Bu for all u ∈U. We will not use a special notation for this type of restriction. • If C :X→Y andV ⊂X then the map C :V →Y defined by Cx:= Cx for x∈V, is called the (domain) restriction of C toV, and it is denoted CV. • If A:X→X andV⊂X is an A-invariant subspace then the map A:V→ V defined by Ax:= Ax for x∈V is called the restriction of A toV, notation AV. These somewhat abstract definitions will be clarified later on in terms of matrix representations. IfX is an n-dimensional space and q ,..., q is a basis then every 1 n vector x∈X has a unique representation of the form x= x q +···+ x q . 1 1 n n n The coefficients of this representation, written as a column vector in R , form the column of x with respect to q ,..., q : 1 n   x 1   . . x= .   . x n T For typographical reasons, we often write this column as (x ,..., x ) , where the ‘T’ 1 n denotes transposition. If q ,..., q is a basis ofX and r ,..., r is a basis ofY and 1 n 1 p C:X→Y is a linear map, the matrix of C with respect to the bases q ,..., q and 1 n18 Mathematical preliminaries r ,..., r is formed by writing next to each other the columns of Cq ,...,Cq with 1 p 1 n respect to the basis r ,..., r . The result will look like 1 p   c ··· c 11 1n  . .  . . (C):= ,   . . c ··· c p1 pn T where (c ,..., c ) is the column of Cq with respect to r ,..., r . We use brack- 1i pi i 1 p ets around C here to emphasize that we are talking about the matrix of the map. We p×n p×n will use the notationR for the set of p× n matrices. Hence (C)∈R . Once the bases are fixed, the map C and its matrix determine each other uniquely. Also, the operations of matrix addition, scalar multiplication, product of matrices correspond to addition of maps, scalar multiplication of a map, composition of maps, respectively. For this reason it is customary to identify maps with matrices, once the bases of the given spaces are given and fixed. The advantage of the matrix formulation over the more abstract linear-map formulation is that it allows for much more explicit calcula- tions. On the other hand, if one works with linear maps, one does not have to specify bases, which sometimes makes the treatment much more elegant and transparent. IfV⊂X then a basis q ,..., q ofX for which q ,..., q is a basis ofV (where 1 n 1 k k= dimV) is called a basis ofX adapted toV. More generally, ifV ,V ,...,V 1 2 r is a chain of subspaces (i.e.,V ⊂V ⊂···⊂V ), then a basis q ,..., q ofX is 1 2 r 1 n said to be adapted to this chain if there exist numbers k ,..., k such that q ,..., q 1 r 1 k i is a basis ofV for i= 1,..., r. Finally, ifV ,...,V are subspaces ofX such that i 1 r X=V ⊕V ⊕···⊕V , we say that a basis q ,..., q is adapted toV ,...,V if 1 2 r 1 n 1 r there exist numbers k ,..., k such that k = 1, k = n+ 1 and q ,..., q 1 r+1 1 r+1 k k −1 i i+1 is a basis ofV for i= 1,..., r. i We illustrate the use of matrix representations for the restriction operations intro- duced earlier in this section. • If B : U → X is a linear map satisfying im B ⊂ V, we choose a basis q ,..., q ofX adapted toV. Let p ,..., p be a basis ofU. Then the 1 n 1 m matrix representation of B with respect to the chosen bases is of the form  B 1 (B)= , (2.4) 0 k×m where B ∈R (k:= dimV). This particular form is a consequence of the 1 condition im B⊂V. The matrix of the codomain restriction B:U→V with respect to the bases p ,..., p and q ,..., q is B . 1 m 1 k 1 • Let C : X→ Y andV ⊂ X. Let q ,..., q be a basis ofX adapted toV. 1 n Furthermore, let r ,..., r be a basis ofY. The matrix of C with respect to 1 p p×k q ,..., q and r ,..., r is (C) = (C C ), where C ∈ R and C ∈ 1 n 1 p 1 2 1 2 p×(n−k) R . The matrix of CV with respect to these bases is C . 1 • Let A:X→X,V⊂X such that AV⊂V. Let q ,..., q be a basis ofX 1 nInner product spaces 19 adapted toV. The matrix of A with respect to this basis is  A A 11 12 (A)= . (2.5) 0 A 22 The property A = 0 is a consequence of the A-invariance ofV. The matrix 21 of AV is A . 11 2.3 Inner product spaces We assume that the reader is familiar with the concept of inner product. A linear space over the fieldR with a real inner product is called a real inner product space.A linear space over the fieldC with a complex inner product is called a complex inner product space. The most commonly used real inner product space is the linear space n T R with the inner product (x, y):= x y. The most commonly used complex inner n ∗ product space is the linear spaceC with the inner product (x, y):= x y (here, ‘∗’ ∗ T denotes the conjugate transposition, i.e. x =¯ x ). IfX is a (real or complex) inner product space and ifV is a subspace ofX, then ⊥ V will denote the orthogonal complement ofV. It is easy to see that for any pair of subspacesV,W ofX the following equality holds: ⊥ ⊥ ⊥ (V∩W) =V +W LetX andY be (real or complex) inner product spaces with inner products(, ) X ∗ and(, ) , respectively. If C : X→Y is a map, then the adjoint C : Y→X of Y C is the map defined by ∗ (x, C y) = (Cx, y) (2.6) X Y for all x inX and y inY. It can easily be seen that there exists a unique map satisfying these properties. IfX is a (real or complex) inner product space and if A:X→X is a map, then it can be shown by direct verification that the following holds: −1 ⊥ ∗ ⊥ (A V) = A V . ⊥ ∗ It is also easy to verify that ifV is A-invariant, thenV is A -invariant. IfX andY are real inner product spaces, if q ,..., q and r ,..., r are or- 1 n 1 p thonormal bases ofX andY, respectively, and if (C) is the matrix of C with respect ∗ T to these bases, then the matrix of the adjoint map C is equal to the transposed (C) of (C). Indeed, if x and y denote the columns of x and y, respectively, with respect to the given bases, then (2.6) is equivalent to T ∗ T T x (C )y= x (C) y n p for all x inR and y inR . In the same way, one can show that ifX andY are complex inner product spaces, if q ,..., q and r ,..., r are orthonormal bases ofX andY, respectively and if 1 n 1 p