Common Terms Used in System Identification Model Validation is the process of gaining confidence in a model Essentially this is achieved by"twisting and turning" the model to scrutinize all aspects of it. Of particular importance is the model's ability to reproduce the behavior of the Validation Data sets. Thus it is important to inspect the properties of the residuals from the model when applied to the validation Data 1-5
Common Terms Used in System Identification 1-5 • Model Validation is the process of gaining confidence in a model. Essentially this is achieved by “twisting and turning” the model to scrutinize all aspects of it. Of particular importance is the model’s ability to reproduce the behavior of the Validation Data sets. Thus it is important to inspect the properties of the residuals from the model when applied to the Validation Data
1 The System Identification Problem Basic Information About Dynamic Models System Identification is about building Dynamic Models. Some knowledge about such models is therefore necessary for successful use of the toolbox. The topic is treated in several places in Chapter 3, Tutorial. " Also, there is a wide range of textbooks available for introductory and in-depth studies. For basic use of the toolbox, it is sufficient to have quite superficial insights about dynamic models. This section describes such a basic level of knowledge The signals Models describe relationships between measured signals. It is convenient to distinguish between input signals and output signals. The outputs are then partly determined by the inputs. Think for example of an airplane where the inputs would be the different control surfaces, ailerons, elevators, and the like while the outputs would be the airplanes orientation and position. In most cases, the outputs are also affected by more signals than the measured inputs In the airplane example it would be wind gusts and turbulence effects. Such unmeasured inputs"will be called disturbance signals or noise. If we denote inputs, outputs, and disturbances by u, y, and e, respectively, the relationship can be depicted in the following figure Figure 1-1: Input Signals u, Output Signals y, and Disturbances e All these signals are functions of time, and the value of the input at time t will be denoted by u(t). Often, in the identification context, only discrete-time points are considered, since the measurement equipment typically records the signa just at discrete-time instants, often equally spread in time with a sampling interval of Ttime units. The modeling problem is then to describe how the three signals relate to each other. 1-6
1 The System Identification Problem 1-6 Basic Information About Dynamic Models System Identification is about building Dynamic Models. Some knowledge about such models is therefore necessary for successful use of the toolbox. The topic is treated in several places in Chapter 3, “Tutorial.” Also, there is a wide range of textbooks available for introductory and in-depth studies. For basic use of the toolbox, it is sufficient to have quite superficial insights about dynamic models. This section describes such a basic level of knowledge. The Signals Models describe relationships between measured signals. It is convenient to distinguish between input signals and output signals. The outputs are then partly determined by the inputs. Think for example of an airplane where the inputs would be the different control surfaces, ailerons, elevators, and the like, while the outputs would be the airplane’s orientation and position. In most cases, the outputs are also affected by more signals than the measured inputs. In the airplane example it would be wind gusts and turbulence effects. Such ‘‘unmeasured inputs’’ will be called disturbance signals or noise. If we denote inputs, outputs, and disturbances by u, y, and e, respectively, the relationship can be depicted in the following figure. Figure 1-1: Input Signals u, Output Signals y, and Disturbances e All these signals are functions of time, and the value of the input at time t will be denoted by u(t). Often, in the identification context, only discrete-time points are considered, since the measurement equipment typically records the signals just at discrete-time instants, often equally spread in time with a sampling interval of T time units. The modeling problem is then to describe how the three signals relate to each other. y e u
Basic Information About Dynamic Models The Basic Dynamic Model The basic relationship is the linear difference equation. An example of such an equation is the following one H(D)-1.5yt-)+0.7yt-27=0.9u(t-2)+0.5u(t-3T (ARX Such a relationship tells us, for example, how to compute the output y(t)if the input is known and the disturbance can be ignored HD)=1.5yt--0.7yt-2)+0.9u(t-2T)+0.5u(t-3 The output at time tis thus computed as a linear combination of past outputs and past inputs. It follows, for example, that the output at time t depend the input signal at many previous time instants. This is what the word dynamic refers to The identification problem is then to use measurements of u and y to figure out: The coefficients in this equation(i.e,-1.5, 0.7, etc. How many delayed outputs to use in the description(two in the example y(tTand y(t-2D)) The time delay in the system is(2Tin the example: you see from the second equation that it takes 2Ttime units before a change in u will affect y How many delayed inputs to use(two in the example: u(t-2T)and ult-3TD The number of delayed inputs and outputs are usually referred to as the model order(s) Variants of Model Descriptions The model given above is called an ARX model. There are a handful of variants of this model known as Output-Error(OE)models, ARMAX models, FIR models, and Box-Jenkins(BJ)models. These are described later on in the manuaL. at a basic level it is sufficient to think of them as variants of the arx model allowing also a characterization of the properties of the disturbances Linear state-space models are also easy to work with The essential structure variable is just a scalar: the model order. This gives just one knob to turn when searching for a suitable model description. See below. General linear models can be described symbolically by u+he 1-7
Basic Information About Dynamic Models 1-7 The Basic Dynamic Model The basic relationship is the linear difference equation. An example of such an equation is the following one. Such a relationship tells us, for example, how to compute the output y(t) if the input is known and the disturbance can be ignored: The output at time t is thus computed as a linear combination of past outputs and past inputs. It follows, for example, that the output at time t depends on the input signal at many previous time instants. This is what the word dynamic refers to. The identification problem is then to use measurements of u and y to figure out: • The coefficients in this equation (i.e., -1.5, 0.7, etc.). • How many delayed outputs to use in the description (two in the example: y(t-T) and y(t-2T)). • The time delay in the system is (2T in the example: you see from the second equation that it takes 2T time units before a change in u will affect y). • How many delayed inputs to use (two in the example: u(t-2T) and u(t-3T)). The number of delayed inputs and outputs are usually referred to as the model order(s). Variants of Model Descriptions The model given above is called an ARX model. There are a handful of variants of this model known as Output-Error (OE) models, ARMAX models, FIR models, and Box-Jenkins (BJ) models. These are described later on in the manual. At a basic level it is sufficient to think of them as variants of the ARX model allowing also a characterization of the properties of the disturbances e. Linear state-space models are also easy to work with. The essential structure variable is just a scalar: the model order. This gives just one knob to turn when searching for a suitable model description. See below. General linear models can be described symbolically by y=Gu+He y t( ) – 1.5 yt T ( ) – + 0.7 y t( ) – 2T = 0.9u t( ) – 2T + 0.5u t( ) – 3T ( ) ARX y t( ) = 1.5 yt T ( ) – – 0.7 y t( ) – 2T + + 0.9u t( ) – 2T 0.5u t( ) – 3T
1 The System Identification Problem which says that the measured output y(t) is a sum of one contribution that noise He. The symbol G then denotes the dynamic properties of the syslem o comes from the measured input u(t)and one contribution that comes from th that is, how the output is formed from the input For linear systems it is called the transfer function from input to output. The symbol Refers to the noise properties, and is called the disturbance model. It describes how the disturbances at the output are formed from some standardized noise source State-space models are common representations of dynamical models. They describe the same type of linear difference relationship between the inputs and the outputs as in the arx model, but they are rearranged so that only one delay is used in the expressions. To achieve this, some extra variables, the state variables, are introduced. They are not measured but can be reconstructed from the measured input-output data. This is especially useful when there are several output signals, i.e., when y(t) is a vector. Chapter 3 "Tutorial", gives more details about this. For basic use of the toolbox it is sufficient to know that the order of the state-space model relates to the number of delayed inputs and outputs used in the corresponding linear difference equation. The state-space representation looks like x(t+1)=Ax(t)+Bu(t+Ke(t) y(t=Cx(t)+Du(t)+e(t) Here x(t)is the vector of state variables The model order is the dimension of this vector. The matrix K determines the disturbance properties. Notice that if K=0, then the noise source e(t)affects only the output, and no specific model of the noise properties is built. This corresponds to H= 1 in the general description above, and is usually referred to as an Output-Error model Notice lso that D=0 means that there is no direct influence from u(t)to y(t). Thus the effect of the input on the output all passes via x(t) and will thus be delayed at least one sample. The first value of the state variable vector x(O)reflects the initial conditions for the system at the beginning of the data record. When dealing with models in state-space form, a typical option is whether to estimate D, K, and x(O or to let them b How to Interpret the Noise Source In many cases of system identification, the effects of the noise on the output are insignificant compared to those of the input. With good signal-to-noise ratios (SNR), it is less important to have an accurate disturbance model 1-8
1 The System Identification Problem 1-8 which says that the measured output y(t) is a sum of one contribution that comes from the measured input u(t) and one contribution that comes from the noise He. The symbol G then denotes the dynamic properties of the system, that is, how the output is formed from the input. For linear systems it is called the transfer function from input to output. The symbol H refers to the noise properties, and is called the disturbance model. It describes how the disturbances at the output are formed from some standardized noise source e(t). State-space models are common representations of dynamical models. They describe the same type of linear difference relationship between the inputs and the outputs as in the ARX model, but they are rearranged so that only one delay is used in the expressions. To achieve this, some extra variables, the state variables, are introduced. They are not measured, but can be reconstructed from the measured input-output data. This is especially useful when there are several output signals, i.e., when y(t) is a vector. Chapter 3, “Tutorial”, gives more details about this. For basic use of the toolbox it is sufficient to know that the order of the state-space model relates to the number of delayed inputs and outputs used in the corresponding linear difference equation. The state-space representation looks like x(t+1)=Ax(t)+Bu(t)+Ke(t) y(t)=Cx(t)+Du(t)+e(t) Here x(t) is the vector of state variables. The model order is the dimension of this vector. The matrix K determines the disturbance properties. Notice that if K = 0, then the noise source e(t) affects only the output, and no specific model of the noise properties is built. This corresponds to H = 1 in the general description above, and is usually referred to as an Output-Error model. Notice also that D = 0 means that there is no direct influence from u(t) to y(t). Thus the effect of the input on the output all passes via x(t) and will thus be delayed at least one sample. The first value of the state variable vector x(0) reflects the initial conditions for the system at the beginning of the data record. When dealing with models in state-space form, a typical option is whether to estimate D, K, and x(0) or to let them be zero. How to Interpret the Noise Source In many cases of system identification, the effects of the noise on the output are insignificant compared to those of the input. With good signal-to-noise ratios (SNR), it is less important to have an accurate disturbance model
Basic Information About Dynamic Models Nevertheless it is important to understand the role of the disturbances and the noise source e(t), whether it appears in the arx model or in the general descriptions given above There are three aspects of the disturbances that should be stressed Understanding white noise Using the noise source when working with the model The discussed one by on How can we understand white noise? From a formal point of view, the noise source e will normally be regarded as white noise. This means that it is entirely unpredictable. In other words, it is impossible to guess the value of e(t)no matter how accurately we have measured past data up to time t-1 How can we interpret the noise source? The actual disturbance contribution to the output, He has real significance. It contains all the influences on the measured y, known and unknown, that are not contained in the input u. It explains and captures the fact that even if an experiment is repeated with the same input, the output signal will typically be somewhat different. However, the noise source e need not have a physical significance. In the airplane mple mentioned earlier, the disturbance effects are wind gusts and turbulence. Describing these as arising from a white noise source via a transfer function H, is just a convenient way of capturing their character How can we deal with the noise source when using the model? If the model is used just for simulation, i.e., the responses to various inputs are to be studied, then the disturbance model plays no immediate role. Since the noise source e(t) for new data will be unknown it is taken as zero in the simulations so as to study the effect of the input alone(a noise-free simulation). Making another simulation with e being arbitrary white noise will reveal how reliable the result of the simulation is, but it will not give a more accurate simulation result for the actual systems response. It is a different thing when the model is used for prediction: Predicting future outputs from inputs and previously measured outputs, means that also future disturbance contributions have to be predicted A known, or estimated, correlation structure(which really is the disturbance model) for the disturbances, will allow predictions of future disturbances based on the previously measured values 19
Basic Information About Dynamic Models 1-9 Nevertheless it is important to understand the role of the disturbances and the noise source e(t), whether it appears in the ARX model or in the general descriptions given above. There are three aspects of the disturbances that should be stressed: • Understanding white noise • Interpreting the noise source • Using the noise source when working with the model These aspects are discussed one by one. How can we understand white noise? From a formal point of view, the noise source e will normally be regarded as white noise. This means that it is entirely unpredictable. In other words, it is impossible to guess the value of e(t) no matter how accurately we have measured past data up to time t-1. How can we interpret the noise source? The actual disturbance contribution to the output, H e, has real significance. It contains all the influences on the measured y, known and unknown, that are not contained in the input u. It explains and captures the fact that even if an experiment is repeated with the same input, the output signal will typically be somewhat different. However, the noise source e need not have a physical significance. In the airplane example mentioned earlier, the disturbance effects are wind gusts and turbulence. Describing these as arising from a white noise source via a transfer function H, is just a convenient way of capturing their character. How can we deal with the noise source when using the model? If the model is used just for simulation, i.e., the responses to various inputs are to be studied, then the disturbance model plays no immediate role. Since the noise source e(t) for new data will be unknown, it is taken as zero in the simulations, so as to study the effect of the input alone (a noise-free simulation). Making another simulation with e being arbitrary white noise will reveal how reliable the result of the simulation is, but it will not give a more accurate simulation result for the actual system’s response. It is a different thing when the model is used for prediction: Predicting future outputs from inputs and previously measured outputs, means that also future disturbance contributions have to be predicted. A known, or estimated, correlation structure (which really is the disturbance model) for the disturbances, will allow predictions of future disturbances, based on the previously measured values